\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 212, pp. 1--21.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2013/212\hfil Branches for ground states] {Asymptotic behaviour of branches for ground states of elliptic systems} \author[V. Bobkov, Y. Il'yasov \hfil EJDE-2013/212\hfilneg] {Vladimir Bobkov, Yavdat Il'yasov} % in alphabetical order \address{Yavdat Il'yasov \newline Institute of Mathematics of RAS, Ufa, Russia} \email{ilyasov02@gmail.com} \address{Vladimir Bobkov \newline Institute of Mathematics of RAS, Ufa, Russia \newline Ufa State Aviation Technical University, Ufa, Russia} \email{bobkovve@gmail.com} \thanks{Submitted August 30, 2013. Published September 25, 2013.} \subjclass[2000]{35J50, 35J55, 35J60, 35J70, 35R05} \keywords{System of elliptic equations; p-laplacian; indefinite nonlinearity; \hfill\break\indent Nehari manifold; fibering method} \begin{abstract} We consider the behaviour of solutions to a system of homogeneous equations with indefinite nonlinearity depending on two parameters $(\lambda, \mu)$. Using spectral analysis a critical point $(\lambda^*, \mu^*)$ of the Nehari manifolds and fibering methods is introduced. We study a branch of a ground state and its asymptotic behaviour, including the blow-up phenomenon at $(\lambda^*, \mu^*)$. The differences in the behaviour of similar branches of solutions for the prototype scalar equations are discussed. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article, we discuss the ground state branch to the following system of equations of variational form $$\label{D} \begin{gathered} -\Delta_p u = \lambda |u|^{p-2} u + \alpha f(x) |u|^{\alpha-2} |v|^{\beta} u , \quad x \in \Omega, \\ -\Delta_q v = \mu |v|^{q-2} v + \beta f(x) |u|^{\alpha} |v|^{\beta-2} v, \quad x \in \Omega, \\ u|_{\partial \Omega} = v|_{\partial \Omega} = 0, \end{gathered}$$ where $\Omega \subset \mathbb{R}^n$, $n\geq 1$ is a bounded domain with $C^1$-boundary $\partial \Omega$, $\lambda, \mu \in \mathbb{R}$, $1< p< +\infty$, $1< q < +\infty$ and $$\label{Sob} \alpha, \beta > 0, \quad \frac{\alpha}{p} +\frac{\beta}{q} > 1, \quad \frac{\alpha}{p^*} +\frac{\beta}{q^*} < 1.$$ Here $p^*$ and $q^*$ are the standard critical Sobolev exponents. We suppose $f \in L^\infty(\Omega)$ and that the function $f$ may change sign on $\Omega$; i.e., the problem \eqref{D} has indefinite nonlinearity (cf. \cite{Alam, BCDN}). Hereinafter we will always assume $f \not\equiv 0$ in $\Omega$. The problem \eqref{D} is actually a generalization of the problem with a single equation $$\label{pb} \begin{gathered} -\Delta_p w = \lambda |w|^{p-2}w+ f(x)|w|^{\gamma-2}w, \quad x \in \Omega, \\ w|_{\partial \Omega}=0, \end{gathered}$$ where $\gamma = \alpha + \beta$ and $p < \gamma 0$ (see Remark 10.1 in Section 10). However, in the present paper we show, on the example of \eqref{pb} and \eqref{D} (in the case $p=q$, $\lambda=\mu$), that there is an essential distinction in geometrical structure for the sets of positive solutions of scalar elliptic equations and its corresponding vector generalizations. To this end we study the ground states of \eqref{D} and its behaviour with respect to parameter $\lambda=\mu$ and compare it with known results on \eqref{pb}. The distinctions can be seen in Figs. 2 and 3 below, where under different assumptions about the function $f$ a level lines of the corresponding energy functional $\mathcal{E}(\lambda,\mu)=E_{\lambda,\mu}(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ are shown for \eqref{pb} and \eqref{D}, respectively. Note that the results for $p=q$ are a corollary of our main Theorems \ref{thm1}--\ref{thm4}, where the common cases are considered. The ground state of \eqref{D} will be obtained using Nehari manifolds method \cite{Neh, Szulkin} with its fibering approach \cite{Poh, Poh2}. It should be noted that the direct application of these methods is impossible, since sufficient conditions to this end have to be satisfied. To overcome these difficulties we follow \cite{ilyas, ilst, ilIZV}, where in the investigation of one-parameter problems it has been proposed to find critical values of parameter $\lambda$, which separate intervals where sufficient conditions of Nehari manifold and fibering method are satisfied. As in the investigation of the one-parameter problem \eqref{pb} (see \cite{ilyas, ilIZV}) it can be shown that one of the critical points of \eqref{D} is determined by the first eigenvalues $\lambda_1$ and $\mu_1$ of the Dirichlet operators $-\Delta_p$ and $-\Delta_q$, respectively. This point $(\lambda_1, \mu_1)$ divides the plane $(\lambda, \mu)$ into four quadrants I-IV (see Fig.1). Actually the existence of ground states in the positive part of quadrant I follows from \cite{Bozhkov}. However, our research in this quadrant provide a previously unknown properties of the solutions. The main novelty in the present paper is the investigation of the existence and nonexistence of ground states in quadrant IV, that is why special attention is paid to investigation along the line $(\sigma \lambda_1, \sigma \mu_1)$, $\sigma \in \mathbb{R}$ (see Figure \ref{fig1}). \begin{figure}[ht] \begin{center} \includegraphics[width=0.6\textwidth]{fig1} % Fig_x1.eps \end{center} \caption{The plane $(\lambda, \mu)$} \label{fig1} \end{figure} The article is organized as follows. In Section 2 we present the main results of the paper. In Section 3 we give some preliminaries on the Nehari manifold and fibering method of \eqref{D}. In Section 4 we study critical values of Nehari manifold and fibering method. In Section 5 we prove the existence of ground states of \eqref{D} in quadrant I. In Section 6 we discuss the existence of ground states in quadrant IV. In Section 7 we explore a continuity of the ground states with respect to parameters $(\lambda, \mu)$. In Section 8 we study the behaviour of the energy levels of ground state branches at boundaries of domain of their definition. In Section 9 we obtain some blow-up results for the ground state branches. Section 10 is devoted to final remarks and open problems. \section{Main results} Henceforth we will use the short notation $\bar{\lambda} = (\lambda, \mu)$ and $$\Omega^+ :=\{x\in \Omega: f(x)> 0\}, \quad \Omega^0 :=\{x\in \Omega: f(x)=0\}.$$ We mean that a subset $U$ of $\Omega$ is nonempty if the Lebesgue measure of $U$ is nonzero. By $W^{1,p}_0$ and $W^{1,q}_0$ we denote the standard Sobolev spaces on $\Omega$ with the norms $$\|u\|_p := \Big(\int_\Omega |\nabla u|^p \, dx\Big)^{1/p}, \quad \|v\|_q := \Big(\int_\Omega |\nabla v|^q \, dx\Big)^{1/q},$$ respectively. We will use the symbols $(\lambda_1, \varphi_1)$ and $(\mu_1, \psi_1)$ for the first eigenpairs of the operators $-\Delta_p$ and $-\Delta_q$ in $\Omega$ with zero boundary conditions, respectively. It is known that the eigenvalues $\lambda_1,\mu_1$ are positive, simple and isolated, and the corresponding eigenfunctions $\varphi_1, \psi_1$ are positive and can be normalized so that $\|\varphi_1\|_p=1$, $\|\psi_1\|_q=1$ \cite{Anan, DrabekT, lindqvist}. Problem \eqref{D} has a variational form with the energy functional $$E_{\bar{\lambda}}(u,v)=\frac{1}{p}P_{\lambda}(u)+\frac{1}{q}Q_{\mu}(v) - F(u,v),$$ which is well defined on $W := W^{1,p}_0 \times W^{1,q}_0$ under assumption \eqref{Sob}, and where \begin{gather*} P_{\lambda}(u) := \int_\Omega |\nabla u|^p \, dx-\lambda \int_\Omega |u|^p \, dx,\\ Q_{\mu}(v) := \int_\Omega |\nabla v|^q \, dx- \mu \int_\Omega |v|^q \, dx, \\ F(u, v) := \int_\Omega f(x)|u|^{\alpha}|v|^{\beta}\, dx. \end{gather*} We obtain solutions of \eqref{D} using the constrained minimization problem $$\label{eq:c_lambda} n_{\bar{\lambda}} := \inf \{ E_{\bar{\lambda}}(u,v): (u,v) \in \mathcal{N}_{\bar{\lambda}} \},$$ where $\mathcal{N}_{\bar{\lambda}}$ is the Nehari manifold $$\mathcal{N}_{\bar{\lambda}}:=\{(u,v) \in W \setminus \{0\}: P_{\lambda}(u)- \alpha F(u,v)=0, \; Q_{\mu}(v)- \beta F(u,v)=0 \}.$$ In the case $\mathcal{N}_{\bar{\lambda}} = \emptyset$ we assume $n_{\bar{\lambda}} = +\infty$. We say that a weak solution of \eqref{D} is a \textit{ground state} if it provides minimum in \eqref{eq:c_lambda}. Our definition is slightly different from the standard one (see \cite{Col}). In particular, we allow the ground state to be zero when $n_{\bar{\lambda}}=0$. In the article following critical value plays a crucial role: $$\label{mint0} \sigma^* = \inf_{u,v} \Big[ \max \Big\{ \frac{1}{\lambda_1} \frac{\int |\nabla u|^p \, dx}{\int |u|^p \, dx}, \frac{1}{\mu_1} \frac{\int |\nabla v|^q \, dx}{\int |v|^q \, dx} \Big\}: F(u,v) \geq 0\Big].$$ The main properties of $\sigma^*$ are given in the following lemma. \begin{lemma} \label{lemm:sigma} Assume \eqref{Sob} is satisfied, $p,q \in (1, +\infty)$ and $f \in L^{\infty}(\Omega)$. Then \begin{itemize} \item[(I)] $1 \leq \sigma^* < +\infty$; \item[(II)] $1 < \sigma^*$ if and only if $F(\varphi_1, \psi_1) < 0$. \end{itemize} \end{lemma} Let us denote $\lambda^* := \sigma^* \lambda_1$, $\mu^* := \sigma^* \mu_1$ and \begin{gather*} \Sigma_1 := \{ \bar{\lambda} = (\lambda, \mu) \in \mathbb{R}^2: -\infty < \lambda < \lambda_1,\; -\infty < \mu < \mu_1\}, \\ \Sigma^* := \{ \bar{\lambda} = (\lambda, \mu) \in \mathbb{R}^2: \lambda_1 < \lambda < \lambda^*, \; \mu_1 < \mu < \mu^*\}. \end{gather*} In fact, $\Sigma_1$ is the quadrant I, and $\Sigma^*$ is a subset of quadrant IV (see Figure \ref{fig1}). Our first main result is the following. \begin{theorem} \label{thm1} Assume \eqref{Sob} is satisfied, $p,q \in (1, +\infty)$ and $f \in L^{\infty}(\Omega)$. \begin{itemize} \item[(I)] If $\Omega^+\neq \emptyset$, then \eqref{D} possesses a ground state $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ for all $\bar{\lambda} \in \Sigma_1$ such that $E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) > 0$ and $u_{\bar{\lambda}}, v_{\bar{\lambda}}>0$ in $\Omega$; \item[(II)] if $F(\varphi_1, \psi_1) < 0$, then \eqref{D} possesses a ground state $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ for all $\bar{\lambda} \in \Sigma^*$such that $E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) < 0$ and $u_{\bar{\lambda}}, v_{\bar{\lambda}}>0$ in $\Omega$; \end{itemize} \end{theorem} This result can be clarified by obtaining some continuity properties of ground states with respect to $\bar{\lambda}$ and its asymptotic properties at the boundaries of $\Sigma_1$ and $\Sigma^*$. To this end we study levels of the energy functional $E_{\bar{\lambda}}$ on the solutions of \eqref{D} and consider $$\mathcal{E}(\bar{\lambda}) := E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}), \quad \bar{\lambda} \in \mathbb{R}^2.$$ \begin{theorem} \label{thm2} Assume \eqref{Sob} is satisfied, $p,q \in (1, +\infty)$ and $f \in L^{\infty}(\Omega)$. \begin{itemize} \item[(I)] If $\Omega_+ \neq \emptyset$, then \begin{itemize} \item[(a)] the function $\mathcal{E}(\bar{\lambda})$ in $\Sigma_1$ is continuous; \item[(b)] $\mathcal{E}(\bar{\lambda}) \to 0$ as $\lambda \uparrow \lambda_1$ and $\mu \uparrow \mu_1$, \item[(c)] $\mathcal{E}(\bar{\lambda}) \to 0$ as $\bar{\lambda} \to (\lambda_1, \mu_0)$ for any $\mu_0 < \mu_1$, and $\mathcal{E}(\bar{\lambda}) \to 0$ as $\bar{\lambda} \to (\lambda_0, \mu_1)$ for any $\lambda_0 < \lambda_1$; \end{itemize} \item[(II)] If $F(\varphi_1, \psi_1) < 0$, then \begin{itemize} \item[(a)] the function $\mathcal{E}(\bar{\lambda})$ in $\Sigma^*$ is continuous; \item[(b)] $\mathcal{E}(\bar{\lambda}) \to 0$ as $\lambda \downarrow \lambda_1$ and $\mu \downarrow \mu_1$; \item[(c)] $\mathcal{E}(\bar{\lambda}) \to 0$ as $\bar{\lambda} \to (\lambda_1, \mu_0)$ for any $\mu_0 \in (\mu_1, \mu^*)$, and $\mathcal{E}(\bar{\lambda}) \to 0$ as $\bar{\lambda} \to (\lambda_0, \mu_1)$ for any $\lambda_0 \in (\lambda_1, \lambda^*)$; \end{itemize} \item[(III)] If $f(x) \leq 0$, $p, q \geq 2$ and $\max\{p, q\} > 2$, then $\mathcal{E}(\bar{\lambda}) \to -\infty$ as $\bar{\lambda} \to (\lambda^*, \mu^*)$. \end{itemize} \end{theorem} The statements of Theorems \ref{thm1}, \ref{thm2} should be supplemented by the fact that in quadrants II and III, i.e. for $\lambda<\lambda_1$, $\mu>\mu_1$ and $\lambda>\lambda_1$, $\mu<\mu_1$, we have $n_{\bar{\lambda}}=0$ (see Remark \ref{rmk10.2} in Section 10). Our next results deal with the blow-up behaviour of ground states $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ at the boundaries of $\Sigma_1$ and $\Sigma^*$. \begin{theorem}\label{thm3} Assume \eqref{Sob} is satisfied, $p,q \in (1, +\infty)$, $f \in L^{\infty}(\Omega)$ and $\Omega_+ \neq \emptyset$. \begin{enumerate} \item Let $q < \beta$. Then $\|u_{\bar{\lambda}}\|_p \to +\infty$ and $\|v_{\bar{\lambda}}\|_q \to 0$ as $\lambda \uparrow \lambda_1$, $\mu \to \mu_0$ for any $\mu_0 < \mu_1$. \item Let $p < \alpha$. Then $\|u_{\bar{\lambda}}\|_p \to 0$ and $\|v_{\bar{\lambda}}\|_q \to +\infty$ as $\lambda \to \lambda_0$, $\mu \uparrow \mu_1$ for any $\lambda_0 < \lambda_1$. \item Let $p < \alpha$, $q < \beta$, $F(\varphi_1, \psi_1) < 0$ and $(\bar{\lambda}_m)$, $m \in \mathbb{N}$ be a sequence in $\Sigma_1$ such that $\lambda_m \to \lambda_1$ and $\mu_m \to \mu_1$ as $m\to \infty$. Then up to a subsequence one of the following convergences hold: \begin{itemize} \item $\|u_{\bar{\lambda}_m}\|_p \to \infty$ and $\|v_{\bar{\lambda}_m}\|_q \to 0$ as $m\to \infty$, or \item $\|u_{\bar{\lambda}_m}\|_p \to 0$ and $\|v_{\bar{\lambda}_m}\|_q \to \infty$ as $m\to \infty$. \end{itemize} \end{enumerate} \end{theorem} \begin{theorem}\label{thm4} Assume \eqref{Sob} is satisfied, $f \in L^{\infty}(\Omega)$, $f(x) \leq 0$, $p, q \geq 2$ and $\max\{p, q\} > 2$. Then $\|u_{\bar{\lambda}}\|_p \to +\infty$ and $\|v_{\bar{\lambda}}\|_q \to +\infty$ as $\bar{\lambda} \to (\lambda^*, \mu^*)$. \end{theorem} It is relevant to remark that in this theorem and in statement III of Theorem \ref{thm2} the assumption $f(x) \leq 0$ includes the case $f(x) < 0$ in $\Omega$. Notice that for the scalar problem \eqref{pb} in the case $f(x) < 0$ the corresponding ground states $w_\lambda$ cannot blow up at any finite value of parameter $\lambda$ (see Section 10). It is interesting to consider a special case of \eqref{D}, with $p=q$ and $\lambda = \mu$. Note that in this case $\lambda^* = \mu^*$. From Lemma \ref{lemm:sigma} and Theorem \ref{thm1} we have the following result. \begin{corollary}\label{cor1} Assume $p=q$, $\lambda=\mu$, $p<\alpha+\beta 2$, then $\mathcal{E}(\lambda) \to -\infty$ as $\lambda \to \lambda^*$. \end{itemize} \end{corollary} \section{Nehari manifold and fibering method} According to the fibering method \cite{Poh, Poh2} consider \begin{eqnarray*} E_{\bar{\lambda}}(t u, s v)=\frac{t^p}{p}P_{\lambda}(u)+\frac{s^q}{q}Q_{\mu}(v) - t^\alpha s^\beta F(u,v), \quad t,s >0, \end{eqnarray*} for $(u, v) \in W$, and the system of equations $$\label{sys:neh} \begin{gathered} \frac{\partial}{\partial t}E_{\bar{\lambda}}(t u, s v)\equiv t^{p-1} P_{\lambda}(u)- \alpha t^{\alpha -1} s^{\beta} F(u,v)=0, \\ \frac{\partial}{\partial s}E_{\bar{\lambda}}(t u, s v)\equiv s^{q-1}Q_{\mu}(v)- \beta t^{\alpha} s^{\beta-1}F(u,v)=0. \end{gathered}$$ Simple analysis shows that only if $(u,v)$ belongs to one of the following sets \begin{align*} \mathcal{A}:=\{(u,v) \in W: P_{\lambda}(u)>0, \; Q_{\mu}(v)>0, \; F(u,v)>0\}, \\ \mathcal{B}:=\{(u,v) \in W: P_{\lambda}(u)<0, \; Q_{\mu}(v)<0, \; F(u,v)<0\}, \end{align*} the system \eqref{sys:neh} has a unique nontrivial solution $s=s(u,v)$, $t=t(u,v)$ and \begin{gather} \label{fiber_t} t^{p q d} = \frac{\alpha^{\beta - q}}{\beta^{\beta}} \, \frac{|P_{\lambda}(u)|^{q-\beta} |Q_{\mu}(v)|^\beta}{|F(u, v)|^{q}},\\ \label{fiber_s} s^{p q d} = \frac{\beta^{\alpha - p}}{\alpha^{\alpha}} \, \frac{|P_{\lambda}(u)|^{\alpha} |Q_{\mu}(v)|^{p - \alpha}}{|F(u, v)|^{p}}, \end{gather} where we denote $$d := \frac{\alpha}{p} + \frac{\beta}{q} - 1.$$ Substituting these roots to $E_{\bar{\lambda}}(t u, s v)$ we obtain the function $$\label{fiber_J} \mathcal{J}_{\bar{\lambda}}(u, v) := E_{\bar{\lambda}}(t(u,v) u, s(u,v) v) = C \frac{|P_{\lambda}(u)|^{\alpha/(p d)} |Q_{\mu}(v)|^{\beta/(q d)}} {|F(u, v)|^{1/d}}\operatorname{sign}(F(u, v)),$$ where $$C = \Big(\frac{1}{\alpha^{\alpha q} \beta^{\beta p}}\Big)^{1/(p q d)}d.$$ Observe that $\mathcal{J}_{\bar{\lambda}}(u, v)$ is zero-homogeneous and weak lower semicontinuous function on $W\setminus \{0\}$. Consider the Nehari manifold corresponding to \eqref{D}, $\mathcal{N}_{\bar{\lambda}}:=\{(u,v) \in W \setminus \{0\}: P_{\lambda}(u)- \alpha F(u,v)=0, \, Q_{\mu}(v)- \beta F(u,v)=0 \},$ and the Hessian of $E_{\bar{\lambda}}(u,v)$, $$\Gamma_{\bar{\lambda}}(u,v) = \begin{pmatrix} D_{uu}E_{\bar{\lambda}}(u,v) & D_{uv}E_{\bar{\lambda}}(u,v) \\ D_{uv}E_{\bar{\lambda}}(u,v) & D_{vv}E_{\bar{\lambda}}(u,v) \end{pmatrix},$$ where \begin{gather*} D_{uu}E_{\bar{\lambda}}(u,v) = (p-1) P_{\lambda}(u)- \alpha (\alpha -1) F(u,v), \quad D_{uv}E_{\bar{\lambda}}(u,v) = -\alpha \beta F(u,v), \\ D_{vv}E_{\bar{\lambda}}(u,v) = (q-1) Q_{\mu}(v)- \beta (\beta -1) F(u,v)\,. \end{gather*} \begin{lemma}\label{LS1} Let $\bar{\lambda} \in \mathbb{R}^2$ and $(u_0, v_0)$ be a solution of \eqref{eq:c_lambda} such that $$\det \Gamma_{\bar{\lambda}} (u_0, v_0) \neq 0.$$ Then $(u_0, v_0)$ is a critical point of $E_{\bar{\lambda}}(u, v)$; i.e., a weak solution of \eqref{D}. \end{lemma} \begin{proof} Let $\bar{\lambda} \in \mathbb{R}^2$ and $(u_0, v_0)$ be a solution of \eqref{eq:c_lambda}. Then by the Lagrange multiplier rule there exist $\mu_0$, $\mu_1$, $\mu_2$ such that $|\mu_0|+|\mu_1|+|\mu_2|\neq 0$ and \begin{align*} &\mu_0 D_u E_{\bar{\lambda}}(u_0, v_0)(\xi) + \mu_1 [ D_u E_{\bar{\lambda}}(u_0, v_0)(\xi) + D_{uu}E_{\bar{\lambda}}(u_0, v_0)(u_0,\xi) ] \\ &+ \mu_2 D_{uv}E_{\bar{\lambda}}(u_0, v_0)(v_0, \xi) = 0; \\ &\mu_0 D_v E_{\bar{\lambda}}(u_0, v_0)(\zeta) + \mu_2 [ D_{vv}E_{\bar{\lambda}}(u_0, v_0)(\zeta, v_0) + D_{v}E_{\bar{\lambda}}(u_0, v_0)(\zeta) ] \\ &+ \mu_1 D_{uv} E_{\bar{\lambda}}(u_0, v_0)(u_0,\zeta) = 0, \end{align*} for all $\xi \in W_0^{1,p}$ and $\zeta \in W_0^{1,q}$. The proof will be obtained if we show that $\mu_1=\mu_2=0$. Let $\xi=u_0$ and $\zeta=v_0$. Then taking into account that $(u_0, v_0) \in \mathcal{N}_{\bar{\lambda}}$ we obtain \begin{gather*} \mu_1 D_{u}E_{\lambda}(u_0, v_0)(u_0, u_0) + \mu_2 D_{uv}E_{\lambda}(u_0, v_0)(v_0, u_0)=0, \\ \mu_1 D_{uv} E_{\lambda}(u_0, v_0)(v_0, u_0)+ \mu_2 D_{vv}E_{\lambda}(u_0, v_0)(v_0, v_0)=0. \end{gather*} But under the assumption $\det \Gamma_{\lambda} (u_0, v_0) \neq 0$ this is possible if and only if $\mu_1=\mu_2=0$. \end{proof} It is not hard to see that on $\mathcal{N}_{\bar{\lambda}}$ one has $$\Gamma_{\bar{\lambda}}(u,v) = \begin{pmatrix} \alpha (p - \alpha) F(u,v) & -\alpha \beta F(u,v) \\ -\alpha \beta F(u,v) & \beta (q-\beta)F(u,v) \end{pmatrix}$$ Hence $$\label{DDET} \det \Gamma_{\lambda}(u,v) = \alpha \beta p q \big(1 - \frac{\alpha}{p} - \frac{\beta}{q} \big) F^2(u,v).$$ Under assumptions $\alpha, \beta > 0$ and $F(u, v) \neq 0$ it can be highlighted three cases: $$\frac{\alpha}{p} + \frac{\beta}{q} = 1, \quad \frac{\alpha}{p} + \frac{\beta}{q} < 1, \quad \frac{\alpha}{p} + \frac{\beta}{q} > 1,$$ that corresponds to the cases when $\det \Gamma_{\lambda}(u,v)$ is zero, positive and negative, respectively. Note that in the present paper we deal only with the last case with negative determinant $\det \Gamma_{\lambda}(u,v)<0$ that corresponds in the case $p=q$, $\lambda=\mu$ to the super-linear problem \eqref{pb}. \section{On critical values of Nehari manifolds and fibering methods} As it has been noted above we study \eqref{D} using constrained minimization method with Nehari manifold $\mathcal{N}_{\bar{\lambda}}$ as a constraint. However, the application of this method is restricted by the assumption of Lemma \ref{LS1}. The critical value $\sigma^*$ in \eqref{mint0} is introduced to separate the domains in the plane $(\lambda, \mu)$ where this assumption is satisfied. A general approach of finding such kind of values has been introduced in \cite{ilst, ilIZV} and has been developed in \cite{ilconcan, ilrunst, ilegorov} with applications to different problems. However, the direct application of this theory to the system \eqref{D} is complicated. Moreover, the full theoretical introduction to this approach would lead us away from the main aims of the present paper. We leave it to our forthcoming paper. The proof of Lemma \ref{lemm:sigma} will be a consequence of the next three propositions. \begin{proposition} \label{prop:sigma1} $1 \leq \sigma^*<+\infty$. \end{proposition} \begin{proof} The first estimate easily follows from observation $$\sigma^* \geq \inf_{u,v} \Big[ \max \Big\{ \frac{1}{\lambda_1} \frac{\int |\nabla u|^p \, dx}{\int |u|^p \, dx},\, \frac{1}{\mu_1} \frac{\int |\nabla v|^q \, dx}{\int |v|^q \, dx} \Big\} \Big] = 1,$$ which holds for $(\varphi_1, \psi_1)$. The second estimate is also true, since one can find $u \in W_0^{1,p}\setminus \{0\}$ and $v \in W_0^{1,q}\setminus \{0\}$ such that $\text{supp } u \cap \text{supp } v = \emptyset$. Then $F(u,v)=0$ and $$\sigma^* \leq \max \Big\{ \frac{1}{\lambda_1} \frac{\int |\nabla u|^p \, dx} {\int |u|^p \, dx},\, \frac{1}{\mu_1} \frac{\int |\nabla v|^q \, dx}{\int |v|^q \, dx} \Big\} < +\infty.$$ \end{proof} \begin{proposition}\label{exCr} There exists a nonzero minimizer $(u^*, v^*) \in W$ of \eqref{mint0} such that $u^*, v^* \geq 0$ in $\Omega$. \end{proposition} \begin{proof} Let $(u_k, v_k)$ be a minimizing sequence for \eqref{mint0}. We may assume that $\|u_k\| = 1$ and $\|v_k\| = 1$ for all $k \in \mathbb{N}$, since \eqref{mint0} is zero-homogeneous. Hence by the Eberlein-Shmulyan theorem and the Sobolev embedding theorem there exists a subsequence of $(u_k, v_k)$ (which we denote again $(u_k, v_k)$) and $(u^*, v^*) \in W$ such that \begin{gather*} u_k \rightharpoonup u^* \text{ weakly in } W_0^{1,p}, \quad v_k \rightharpoonup v^* \text{ weakly in } W_0^{1,q}, \\ u_k \to u^* \text{ in } L^{r},\, r < p^*, \quad v_k \to v^* \text{ in } L^{r},\, r < q^*. \end{gather*} This implies $F(u^*, v^*) \geq 0$. From Proposition \ref{prop:sigma1} we know that $\sigma^* < c < +\infty$ for some $c > 0$. Consequently \begin{gather*} \int_\Omega |u^*|^p \, dx = \lim_{k \to +\infty} \int_\Omega |u_k|^p \, dx > 1/c > 0, \\ \int_\Omega |v^*|^q \, dx = \lim_{k \to +\infty} \int_\Omega |v_k|^q \, dx > 1/c > 0, \end{gather*} and therefore, $u^*, v^* \not\equiv 0$. By the weak lower semicontinuity we have $$\frac{\int_\Omega |\nabla u^*|^p}{\int_\Omega |u^*|^p} \leq \liminf_{k \to +\infty} \frac{\int_\Omega |\nabla u_k|^p} {\int_\Omega |u_k|^p}, \quad \frac{\int_\Omega |\nabla v^*|^q}{\int_\Omega |v^*|^q} \leq \liminf_{k \to +\infty} \frac{\int_\Omega |\nabla v_k|^q} {\int_\Omega |v_k|^q}.$$ Now arguing by contradiction we conclude that $(u^*, v^*)$ is a minimizer of \eqref{mint0}. Since the functionals in \eqref{mint0} are even, we may assume that $u^*, v^* \geq 0$. \end{proof} \begin{proposition} \label{sigma>1} $\sigma^* > 1$ if and only if $F(\varphi_1, \psi_1) < 0$. \end{proposition} \begin{proof} Assume first $\sigma^* > 1$. Conversely, suppose that $F(\varphi_1, \psi_1) \geq 0$. Then $(\varphi_1, \psi_1)$ is an admissible point for minimization problem \eqref{mint0}. But then $\sigma^* = 1$, which contradicts the assumption $\sigma^* > 1$. Assume now $F(\varphi_1, \psi_1) < 0$. Suppose, contrary to our claim, that $\sigma^* = 1$. By Proposition \ref{exCr} there exists a nonzero minimizer $(u^*, v^*) \in W$ of \eqref{mint0}. Then it follows easily that $$\frac{1}{\lambda_1} \frac{\int |\nabla u^*|^p \, dx}{\int |u^*|^p \, dx} = 1, \quad \frac{1}{\mu_1} \frac{\int |\nabla v^*|^q \, dx}{\int |v^*|^q \, dx} = 1.$$ These equalities are true only if $u^* = \varphi_1$ and $v^* = \psi_1$ up to multipliers, but it is impossible, since $F(\varphi_1, \psi_1) < 0$. \end{proof} \begin{proposition} \label{contr} Assume $F(\varphi_1, \psi_1) < 0$. Let $\lambda<\lambda^*$ and $\mu<\mu^*$. Then for any $(u,v) \in W \setminus \{0\}$ the following implication is true: $$P_\lambda(u) \leq 0,\; Q_\mu(v) \leq 0 \Longrightarrow F(u, v)<0.$$ \end{proposition} \begin{proof} Let $\lambda<\lambda^*$ and $\mu<\mu^*$. On the contrary, suppose that $P_\lambda(u) \leq 0$, $Q_\mu(v) \leq 0$ and $F(u, v) \geq 0$. Then there exist $\lambda_0 \leq \lambda < \lambda^*$ and $\mu_0 \leq \mu < \mu^*$ such that $P_{\lambda_0}(u) = 0$ and $Q_{\mu_0}(v) = 0$. From here it follows $$\frac{1}{\lambda_1} \frac{\int |\nabla u|^p \, dx}{\int |u|^p \, dx} = \frac{\lambda_0}{\lambda_1} < \frac{\lambda^*}{\lambda_1} = \sigma^*, \quad \frac{1}{\mu_1} \frac{\int |\nabla v|^q \, dx}{\int |v|^q \, dx} = \frac{\mu_0}{\mu_1} < \frac{\mu^*}{\mu_1} = \sigma^*.$$ Hence we obtain $$\max \big\{ \frac{1}{\lambda_1} \frac{\int |\nabla u|^p \, dx} {\int |u|^p \, dx}, \frac{1}{\mu_1} \frac{\int |\nabla v|^q \, dx} {\int |v|^q \, dx} \big\} < \sigma^*,$$ which contradicts the definition of $\sigma^*$. \end{proof} For further applications of Lemma \ref{LS1} we need the next result. \begin{corollary}\label{DCor} Assume \eqref{Sob} is satisfied, $p,q \in (1, +\infty)$ and $f \in L^{\infty}(\Omega)$. Let $\lambda<\lambda^*$ and $\mu<\mu^*$. Then $\det \Gamma_{\bar{\lambda}} (u, v) \neq 0$, for any $(u,v) \in \mathcal{N}_{\bar{\lambda}}$. \end{corollary} \begin{proof} By \eqref{DDET} we know that $$\det \Gamma_{\lambda}(u,v) = \alpha \beta d p q F^2(u,v).$$ Hence, the proof will be obtained if we show that $F(u,v)\neq 0$. Suppose, contrary to our claim, that $F(u,v) = 0$ for $(u,v) \in \mathcal{N}_{\bar{\lambda}}$ and $\lambda<\lambda^*$ and $\mu<\mu^*$. Then the constraints of the Nehari manifold entail $P_\lambda(u) = 0$ and $Q_\mu(v) = 0$. However, this contradicts the definition of $\sigma^*$. \end{proof} \section{Solutions of \eqref{D} in $\Sigma_1$} In this section we prove statement (I) of Theorem \ref{thm1}. First we prove the following proposition. \begin{proposition} \label{lem:N+empty} Assume that $\Omega_+ \neq \emptyset$. Then $\mathcal{N}_{\bar{\lambda}} \neq \emptyset$ for all $\bar{\lambda} \in \mathbb{R}^2$. \end{proposition} \begin{proof} Let $\bar{\lambda} \in \mathbb{R}^2$. Consider the first eigenpairs $(\lambda_1(B), \varphi_1(B))$ and $(\mu_1(B), \psi_1(B))$ of operators $-\Delta_p$ and $-\Delta_q$ in a ball $B \subset \Omega_+$ with zero boundary conditions, respectively. Evidently, we can choose $B$ such that $\lambda < \lambda_1(B)$ and $\mu < \mu_1(B)$. Then $$P_\lambda(\varphi_1(B)) > 0, \quad Q_\mu(\psi_1(B)) > 0, \quad F(\varphi_1(B), \psi_1(B)) > 0.$$ Thus $(\varphi_1(B), \psi_1(B)) \in \mathcal{A}$ and $(t \varphi_1(B), s \psi_1(B)) \in \mathcal{N}_{\bar{\lambda}}$, where $t, s$ are given by \eqref{fiber_t} and \eqref{fiber_s}. \end{proof} \begin{lemma} \label{thm:upper} Let $\Omega_+ \neq \emptyset$ and $\bar{\lambda} \in \Sigma_1$. Then problem \eqref{D} has a ground state $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ such that $E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) > 0$ and $u_{\bar{\lambda}}, v_{\bar{\lambda}} > 0$. \end{lemma} \begin{proof} Assume $\Omega_+ \neq \emptyset$ and $\bar{\lambda} \in \Sigma_1$. Let us consider constrained minimization problem \eqref{eq:c_lambda}. By Proposition \ref{lem:N+empty} we know that $\mathcal{N}_{\bar{\lambda}} \neq \emptyset$. This yields $n_{\bar{\lambda}} < +\infty$. Notice that $\mathcal{N}_{\bar{\lambda}} \subset \mathcal{A}$ for $\bar{\lambda} \in \Sigma_1$, since $P_\lambda(u) > 0$, $Q_\mu(v) > 0$ for $(u,v) \in W \setminus \{0\}$ in case $\lambda < \lambda_1$ and $\mu < \mu_1$. Then $$\label{I:F} E_{\bar{\lambda}}(u,v) = \frac{\alpha}{p}F(u,v) + \frac{\beta}{q}F(u,v) - F(u,v) = \Big( \frac{\alpha}{p} + \frac{\beta}{q} - 1 \Big)F(u,v) > 0.$$ for any $(u,v) \in \mathcal{N}_{\bar{\lambda}}$. This implies that $n_{\bar{\lambda}} \geq 0$. Let $(u_n, v_n)$ be a minimizing sequence of \eqref{eq:c_lambda}. Let us verify the boundedness of $(u_n, v_n)$. Suppose, contrary to our claim, that for instance $\|u_n\|_p \to +\infty$. Then $P_\lambda(u_n) \to +\infty$, because for $\lambda < \lambda_1$ we have $$\label{eq:p>0} P_\lambda(u_n) = \|u_n\|_p^p - \lambda \int |u_n|^p \, dx \geq C_0(\lambda) \|u_n\|_p^p,$$ where $C_0(\lambda) = 1 - \lambda / \lambda_1$ if $\lambda \in (0, \lambda_1)$, and $C_0(\lambda) = 1$ if $\lambda \leq 0$. Since $P_{\lambda}(u_n)- \alpha F(u_n, v_n)=0$, it follows $F(u_n, v_n) \to +\infty$. Thus by \eqref{I:F} we have $E_{\bar{\lambda}} (u_n, v_n) \to +\infty$. But it is impossible, since $n_{\bar{\lambda}}< +\infty$. Thus, $(u_n, v_n)$ is bounded. This implies the existence of subsequence of $(u_n, v_n)$ (which we denote again $(u_n, v_n)$) and $(u, v) \in W$ such that \begin{gather*} u_n \rightharpoonup u \text{ weakly in } W_0^{1,p}, \quad v_n \rightharpoonup v \text{ weakly in } W_0^{1,q}, \\ u_n \to u \text{ in } L^{r}, ~r < p^*, \quad v_n \to v \text{ in } L^{r}, ~r < q^*. \end{gather*} Let us show that $u, v \not\equiv 0$. Note first that similar to \eqref{eq:p>0} we have the following estimate $$\label{eq:p<0} P_\lambda(u_n) \leq C_1(\lambda) \|u_n\|_p^p,$$ where $C_1(\lambda) = 1$ if $\lambda > 0$ and $C_1(\lambda) = 1 - \lambda / \lambda_1$ if $\lambda \leq 0$. Further, from assumptions \eqref{Sob} it follows the existence of $p' \in (p, p^*)$ and $q' \in (q, q^*)$ such that $\frac{\alpha}{p'} +\frac{\beta}{q'} = 1$. Hence, applying H\"{o}lder's inequality and the Sobolev embedding theorem we obtain $$\label{eq:f>0} F(u_n, v_n) \leq C_2 \Big( \int |u_n|^{p'} \, dx \Big)^{\alpha/p'} \Big( \int |v_n|^{q'} \, dx \Big)^{\beta/q'} \leq C_3 \|u_n\|_p^{\alpha} \|v_n\|_q^{\beta},$$ where $C_2$ and $C_3$ are some constants in $(0, +\infty)$ which do not depend on $n \in \mathbb{N}$ and $\lambda$, $\mu$. Since $(u_n, v_n) \in \mathcal{N}_{\bar{\lambda}}$, we have $t_n, s_n = 1$. Let $q \geq \beta$. Substituting estimates \eqref{eq:p>0} and \eqref{eq:f>0} to \eqref{fiber_t} we obtain $$1 = \frac{\alpha^{\beta - q}}{\beta^{\beta}} \frac{P_{\lambda}(u_n)^{q-\beta} Q_{\mu}(v_n)^\beta}{F(u_n, v_n)^{q}} \geq \frac{\alpha^{\beta - q}}{\beta^{\beta}} \frac{C_0(\lambda)^{q-\beta} C_0(\mu)^\beta}{C_3^q} \frac{\|u_n\|_p^{q p - \beta p} \|v_n\|_q^{\beta q}} {\|u_n\|_p^{\alpha q} \|v_n\|_q^{\beta q}},$$ and consequently $$\|u_n\|_p^{p q (\frac{\alpha}{p} + \frac{\beta}{q} - 1)} \geq \frac{\alpha^{\beta - q}}{\beta^{\beta}} \frac{C_0(\lambda)^{q-\beta} C_0(\mu)^\beta}{C_3^q} > 0,$$ for all $n \in \mathbb{N}$. Let $q < \beta$. Using \eqref{eq:p<0} instead of \eqref{eq:p>0}, for all $n \in \mathbb{N}$ we derive $$\|u_n\|_p^{p q (\frac{\alpha}{p} + \frac{\beta}{q} - 1)} \geq \frac{\alpha^{\beta - q}}{\beta^{\beta}} \frac{C_1(\lambda)^{q-\beta} C_0(\mu)^\beta}{C_3^q} > 0.$$ At the same time, we have \label{eq:F>0} \begin{aligned} F(u, v) &= \lim_{n \to +\infty} F(u_n, v_n) \\ &= \frac{1}{\alpha} \lim_{n \to +\infty} P_\lambda(u_n)\\ & \geq \frac{C_0(\lambda)}{\alpha} \lim_{n \to +\infty} \|u_n\|_p^p > C_4(\lambda, \mu) >0, \end{aligned} for some constant $C_4(\lambda, \mu)$ which does not depend on $n$. But this is possible if and only if $u, v \not\equiv 0$. From here and using \eqref{fiber_t}, \eqref{fiber_s} we can find $s, t > 0$ such that $(t u, s v) \in \mathcal{N}_{\bar{\lambda}}$. Since $\mathcal{J}_{\bar{\lambda}}(u, v)$ is a weak lower semicontinuous and zero-homogeneous functional, we have $E_{\bar{\lambda}}(t u, s v) = \mathcal{J}_{\bar{\lambda}}(u, v) \leq \liminf_{n \to +\infty} \mathcal{J}_{\bar{\lambda}}(u_n, v_n) = \liminf_{n \to +\infty} E_{\bar{\lambda}}(u_n, v_n) = n_{\bar{\lambda}}.$ By the definition of $n_{\bar{\lambda}}$ here is only equality possible. Thus $(t u, s v)$ is a solution of \eqref{eq:c_lambda}. Applying Lemma \ref{LS1} and Corollary \ref{DCor} we deduce that $(u_{\bar{\lambda}}:=t u, v_{\bar{\lambda}}:=s v)$ is a weak solution of \eqref{D}. Furthermore, $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ is a ground state, since it is a minimizer of \eqref{eq:c_lambda}. Notice that $E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ is even function with respect to both variables. Therefore, we may assume that $u_{\bar{\lambda}}, v_{\bar{\lambda}} \geq 0$. Applying the arguments of \cite{drabek} we obtain that $u_{\bar{\lambda}}, v_{\bar{\lambda}} > 0$ in $\Omega$. Finally, \eqref{eq:F>0} and \eqref{I:F} entail $E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) > 0$. \end{proof} \section{Solutions of \eqref{D} in $\Sigma^*$} In this section we prove statement (II) of Theorem \ref{thm1}. Below we always suppose that $F(\varphi_1, \psi_1) < 0$. \begin{proposition} \label{lemm:E<0} Let $\lambda > \lambda_1$ and $\mu > \mu_1$. Then $\mathcal{N}_{\bar{\lambda}} \neq \emptyset$. Moreover, there is $(u, v) \in \mathcal{N}_{\bar{\lambda}}$ such that $E_{\bar{\lambda}}(u, v) < 0$. \end{proposition} \begin{proof} Consider $(\varphi_1, \psi_1)$. Since $\lambda > \lambda_1$ and $\mu > \mu_1$, we have $P_\lambda(\varphi_1) < 0$, $Q_\lambda(\psi_1) < 0$, and by the assumption $F(\varphi_1, \psi_1) < 0$. Thus $(\varphi_1, \psi_1) \in \mathcal{B}$ and $(t \varphi_1, s \psi_1) \in \mathcal{N}_{\bar{\lambda}}$, where $t, s$ are given by \eqref{fiber_t} and \eqref{fiber_s}. Furthermore by \eqref{fiber_J} we have $E_{\bar{\lambda}}(t \varphi_1, s \psi_1)=\mathcal{J}_{\bar{\lambda}} (\varphi_1, \psi_1)<0$. \end{proof} \begin{lemma}\label{thm:lower} Let $\bar{\lambda} \in \Sigma^*$. Then problem \eqref{D} has a ground state $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ such that $E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) < 0$ and $u_{\bar{\lambda}}, v_{\bar{\lambda}} > 0$. \end{lemma} \begin{proof} Let $\bar{\lambda} \in \Sigma^*$. As above in the proof of Lemma \ref{thm:upper}, we will obtain the ground state by the finding of a minimizer of \eqref{eq:c_lambda}. By Proposition \ref{lemm:E<0} we know that $\mathcal{N}_{\bar{\lambda}} \neq \emptyset$ and $n_{\bar{\lambda}} < 0$. Let $(u_n, v_n) \in \mathcal{N}_{\bar{\lambda}}$ be a minimizing sequence for \eqref{eq:c_lambda}. Then $E_{\bar{\lambda}}(u_n, v_n) < 0$ and by \eqref{sys:neh}, \eqref{fiber_J} it easy follows that $$P_\lambda(u_n) < 0, \quad Q_\mu(v_n) < 0, \quad F(u_n, v_n) < 0.$$ Let us consider $u_n = t_n \hat{u}_n$ and $v_n = s_n \hat{v}_n$, where $\|\hat{u}_n\|_p=1$ and $\|\hat{v}_n\|_q=1$. Then the boundedness of $(\hat{u}_n)$ and $(\hat{v}_n)$ in $W$ implies the existence of $(\hat{u}, \hat{v}) \in W$ and a subsequence of $(\hat{u}_n, \hat{v}_n)$ (which we denote again $(\hat{u}_n, \hat{v}_n)$) such that \begin{gather*} \hat{u}_n \rightharpoonup \hat{u} \text{ weakly in } W_0^{1,p},\quad \hat{v}_n \rightharpoonup \hat{v} \text{ weakly in } W_0^{1,q}, \\ \hat{u}_n \to \hat{u} \text{ in } L^{r}, r < p^*,\quad \hat{v}_n \to \hat{v} \text{ in } L^{r}, r < q^*. \end{gather*} Observe that $$\label{eq:F<0} F(\hat{u}_n, \hat{v}_n) < C_0 < 0,$$ where $C_0$ does not depend on $n \in \mathbb{N}$. Indeed, suppose conversely that $F(\hat{u}, \hat{v}) = \lim_{n \to +\infty} F(\hat{u}_n, \hat{v}_n) = 0$. Note that by the weak lower semicontinuity we have $$P_\lambda(\hat{u}) \leq \liminf_{n \to +\infty} P_\lambda(\hat{u}_n) \leq 0, \quad Q_\lambda(\hat{v}) \leq \liminf_{n \to +\infty} Q_\lambda(\hat{v}_n) \leq 0.$$ Hence applying Proposition \ref{contr} we obtain a contradiction. Let us show that $n_{\bar{\lambda}} > -\infty$. By \eqref{fiber_J} we have $$\label{eq:Elambda} E_{\bar{\lambda}}(t_n \hat{u}_n, s_n \hat{v}_n) = -C \Big( \frac{|P_{\lambda}(\hat{u}_n)|^{\alpha q} |Q_{\mu}(\hat{v}_n)|^{\beta p}}{|F(\hat{u}_n, \hat{v}_n)|^{pq}} \Big)^{1/(p q d)}.$$ Taking into account that $\|\hat{u}_n\|_p=1$ and $\|\hat{v}_n\|_q=1$ we see that $E_{\bar{\lambda}}(t_n \hat{u}_n, s_n \hat{v}_n) \to -\infty$ if and only if $F(\hat{u}_n, \hat{v}_n) \to 0$. By \eqref{eq:F<0} the last is impossible and therefore, $n_{\bar{\lambda}} > -\infty$. Let us prove the boundedness of $(t_n, s_n)$. Assume for example that $t_n \to +\infty$. In view of \eqref{fiber_t} and by \eqref{eq:F<0} it is possible only in case $q < \beta$ when $P_\lambda(\hat{u}_n) \to 0$. However by \eqref{eq:Elambda} this implies that $E_{\bar{\lambda}}(t_n \hat{u}_n, s_n \hat{v}_n) \to 0$, which contradicts $n_{\bar{\lambda}}<0$. By the similar arguments it can be shown that $s_n, t_n \not\to 0$. Thus, $(u_n, v_n)$ is bounded and up to subsequence we have \begin{gather*} u_n \rightharpoonup u \text{ weakly in } W_0^{1,p},\quad v_n \rightharpoonup v \text{ weakly in } W_0^{1,q}, \\ u_n \to u \text{ in } L^{r},\,r < p^*,\quad v_n \to v \text{ in } L^{r}, \, r < q^*. \end{gather*} Since $u, v \not\equiv 0$, by \eqref{fiber_t} and \eqref{fiber_s}, we can find $s, t > 0$ such that $(t u, s v) \in \mathcal{N}_{\bar{\lambda}}$. Since $\mathcal{J}_{\bar{\lambda}}(u, v)$ is a weak lower semicontinuous and zero-homogeneous functional, we have \begin{align*} E_{\bar{\lambda}}(t u, s v) = \mathcal{J}_{\bar{\lambda}}(u, v) \leq \liminf_{n \to +\infty} \mathcal{J}_{\bar{\lambda}}(u_n, v_n) = n_{\bar{\lambda}}. \end{align*} But by the definition of $n_{\bar{\lambda}}$ here is only equality possible. Thus $(u_{\bar{\lambda}}:=t u, v_{\bar{\lambda}}:=s v)$ is a solution of \eqref{eq:c_lambda} and by Lemma \ref{LS1} and Corollary \ref{DCor} it follows that $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ is a weak solution of \eqref{D}. Arguing as in the proof of Lemma \ref{thm:upper} we derive that $u_{\bar{\lambda}}, v_{\bar{\lambda}} > 0$ and $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ is a ground state. Obviously $E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) < 0$. \end{proof} \section{Continuity of the set of ground states} In this section we prove statements (I):(a) and (II):(a) of Theorem \ref{thm2}. Observe first that by the construction of $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ we have $$n_{\bar{\lambda}} = E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) = \mathcal{J}_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}),$$ for $\bar{\lambda} \in \Sigma_1$ and $\Sigma^*$. Therefore to confirm (I):(a) and (II):(a) in Theorem \ref{thm2} it is sufficient to establish the following result. \begin{lemma} The function $\mathcal{J}_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ is continuous in $\Sigma_1$ and $\Sigma^*$. \end{lemma} \begin{proof} Fix $\bar{\lambda}_0$ in $\Sigma_1$ or $\Sigma^*$ and the ball $B$ correspondingly in $\Sigma_1$ or $\Sigma^*$ with the center $\bar{\lambda}_0$. Let $\bar{\lambda} \in B$. Denote $\Delta \lambda := \lambda - \lambda_0$, $\Delta \mu := \mu - \mu_0$. The proof of the lemma will follows from the next proposition. \end{proof} \begin{proposition} For sufficiently small $|\bar{\lambda} - \bar{\lambda}_0|$ the following inequalities are satisfied \label{eq:j-j} \begin{aligned} &-\Delta \lambda \frac{1}{p} G_p(u_{\bar{\lambda}}) - \Delta \mu \frac{1}{q} G_q (v_{\bar{\lambda}}) + r_1(\Delta \lambda, \Delta \mu) \\ &\leq \mathcal{J}_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) - \mathcal{J}_{\bar{\lambda}_0}(u_{\bar{\lambda}_0}, v_{\bar{\lambda}_0}) \\ &\leq -\Delta \lambda \frac{1}{p} G_p(u_{\bar{\lambda}_0}) - \Delta \mu \frac{1}{q} G_q(v_{\bar{\lambda}_0}) + r_2(\Delta \lambda, \Delta \mu), \end{aligned} where $r_i(\Delta \lambda, \Delta \mu)= o(|\bar{\lambda}-\bar{\lambda}_0|)$, $i=1,2$; i.e., $\frac{r_i(\Delta \lambda, \Delta \mu)}{|\bar{\lambda}-\bar{\lambda}_0| }\to 0$ as $|\bar{\lambda}-\bar{\lambda}_0| \to 0$ uniformly on $B$, and $$G_p(u) := \int_\Omega |u|^p \, dx, \quad G_q(v) := \int_\Omega |v|^q \, dx, \quad (u,v) \in W.$$ \end{proposition} \begin{proof} Note first that $$\label{eq:j>j} \mathcal{J}_{\bar{\lambda}}(u_{\bar{\lambda}_0}, v_{\bar{\lambda}_0}) \geq \mathcal{J}_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}),$$ since $\mathcal{J}_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) =n_{\bar{\lambda}}$. Furthermore, by \eqref{fiber_J} $$\label{eq:j} \mathcal{J}_{\bar{\lambda}}(u_{\bar{\lambda}_0}, v_{\bar{\lambda}_0}) = C \frac{(P_{\lambda_0}(u_{\bar{\lambda}_0}) -\Delta \lambda G_p(u_{\bar{\lambda}_0}))^{\alpha/(p d)} (Q_{\mu_0}(v_{\bar{\lambda}_0}) - \Delta \mu G_q(v_{\bar{\lambda}_0}))^{\beta/(q d)}}{F(u_{\bar{\lambda}_0}, v_{\bar{\lambda}_0})^{1/d}}.$$ Using Taylor's theorem with Lagrange form of the remainder we obtain \label{eq:q_exp} \begin{aligned} &(Q_{\mu_0}(v_{\bar{\lambda}_0})- \Delta \mu G_q(v_{\bar{\lambda}_0})) ^{\beta/(q d)} \\ &= Q_{\mu_0}(v_{\bar{\lambda}_0})^{\beta/(q d)} - \Delta \mu \frac{\beta}{q d} Q_{\mu_0}(v_{\bar{\lambda}_0})^{\frac{\beta}{q d} - 1} G_q(v_{\bar{\lambda}_0}) \\ &\quad - \frac{(\Delta \mu)^2}{2!} \frac{\beta}{q d} \big( \frac{\beta}{q d} - 1 \big)(Q_{\mu_0}(v_{\bar{\lambda}_0}) - \Delta \mu \theta G_q(v_{\bar{\lambda}_0}))^{\frac{\beta}{q d}-2} G_q(v_{\bar{\lambda}_0})^2, \end{aligned} where $\theta \in (0,1)$. Similar formula is true for $(P_{\lambda_0}(u_{\bar{\lambda}_0}) -\Delta \lambda G_p(u_{\bar{\lambda}_0}))^{\alpha/(p d)}$. Substituting these in \eqref{eq:j>j} and using \eqref{fiber_t}, \eqref{fiber_s} with $t_{\bar{\lambda}}, s_{\bar{\lambda}} = 1$ we obtain $$\mathcal{J}_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) - \mathcal{J}_{\bar{\lambda}_0}(u_{\bar{\lambda}_0}, v_{\bar{\lambda}_0}) \leq -\Delta \lambda \frac{1}{p} G_p(u_{\bar{\lambda}_0}) - \Delta \mu \frac{1}{q} G_q(v_{\bar{\lambda}_0})+ r_2(\Delta \lambda, \Delta \mu),$$ where $r_2(\Delta \lambda, \Delta \mu)$ is a sum of the reminder terms which orders with respect to $|\bar{\lambda} - \bar{\lambda}_0|$ are great or equal two. Thus we obtain the second inequality in \eqref{eq:j-j}. The first one is obtained by the same way. To complete the proof of the proposition it remains to show that $r_i(\Delta \lambda, \Delta \mu)= o(|\bar{\lambda}-\bar{\lambda}_0|)$, $i=1,2$ uniformly on $B$. To this end (see \eqref{eq:j} and \eqref{eq:q_exp}) it is sufficient to show that $\|u_{\bar{\lambda}}\|_p$, $\|v_{\bar{\lambda}}\|_q$ are uniformly bounded on $B$ and $$\label{eq:fpq} |F(u_{\bar{\lambda}}, v_{\bar{\lambda}})| \geq c_0 > 0, \quad |P_\lambda(u_{\bar{\lambda}})| \geq c_0 > 0, \quad |Q_\mu(v_{\bar{\lambda}})| \geq c_0 > 0,$$ with constant $c_0$ which does not depend on $\bar{\lambda} \in B$. First consider the case $B \subset \Sigma_1$. Let us show the boundedness of $\|u_{\bar{\lambda}}\|_p$ and $\|v_{\bar{\lambda}}\|_q$. Suppose, contrary to our claim, that there is a sequence $\bar{\lambda}_m$ such that $\|u_{\bar{\lambda}_m}\|_p \to +\infty$ as $\bar{\lambda}_m \to \bar{\lambda}$ for some $\bar{\lambda} \in B$. Arguing as in the proof of Theorem \ref{thm1} (see \eqref{eq:p>0}) we obtain that in this case $P_{\lambda_m}(u_{\bar{\lambda}_m}) \to +\infty$ and $E_{\bar{\lambda}_m}(u_{\bar{\lambda}_m}) \to +\infty$ as $m \to +\infty$. But the last is impossible, since $E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ is uniformly bounded in $B$. Indeed, consider the first eigenfunctions $\varphi_1(U)$ and $\psi_1(U)$ of operators $-\Delta_p$ and $-\Delta_q$ with zero boundary conditions on a smooth subset $U \subset \Omega_+$. Then for all $\bar{\lambda} \in B$ we have $$n_{\bar{\lambda}} \leq \mathcal{J}_{\bar{\lambda}}(\varphi_1(U),\psi_1(U)) < c_3 < +\infty,$$ where $c_3$ does not depend on $\bar{\lambda} \in B$. By \eqref{eq:F>0} we have \begin{equation*} F(u_{\bar{\lambda}}, v_{\bar{\lambda}}) \geq C_4(\lambda, \mu) >0, \quad \bar{\lambda} \in B, \end{equation*} where by the construction $C_4(\lambda, \mu)$ is continuous nonzero function on the compact set $B$. This implies the first estimate in \eqref{eq:fpq}. At the same time we have the Nehari constraints $P_\lambda(u_{\bar{\lambda}}) = \alpha F(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ and $Q_\mu(v_{\bar{\lambda}}) = \beta F(u_{\bar{\lambda}}, v_{\bar{\lambda}})$, which imply the last two estimates in \eqref{eq:fpq}. Now consider the case $B \subset \Sigma^*$. Arguing as in \eqref{eq:F<0} we obtain the first inequality in \eqref{eq:fpq}. Then again using Nehari constraints we obtain the last two estimates in \eqref{eq:fpq}. The boundedness of $\|u_{\bar{\lambda}}\|_p$ and $\|v_{\bar{\lambda}}\|_q$ is shown similar to the proof of Lemma \ref{thm:lower} using Proposition \ref{contr}. \end{proof} \begin{corollary}\label{corD} Assume \eqref{Sob} is satisfied and $p,q \in (1, +\infty)$, $f \in L^{\infty}(\Omega)$. Let $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ be a ground state in $\Sigma_1$ and $\Sigma^*$. Then the function $\mathcal{E}(\bar{\lambda}) = E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ is differentiable at any point $\bar{\lambda} \in \Sigma_1$ and $\Sigma^*$. Furthermore $$\frac{\partial}{\partial \lambda} E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) = -\frac{1}{p} \int_{\Omega} |u_{\bar{\lambda}}|^p \, dx, \quad \frac{\partial}{\partial \mu} E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) = -\frac{1}{q} \int_{\Omega} |v_{\bar{\lambda}}|^q \, dx.$$ \end{corollary} \begin{proof} Let $\bar{\lambda} \in \Sigma_1$ or $\Sigma^*$. Consider the sequence $\bar{\lambda}_i \to \bar{\lambda}$. From the continuity of $E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ it follows that $(E_{\bar{\lambda}_i}(u_{\bar{\lambda}_i}, v_{\bar{\lambda}_i}))$ is uniformly bounded. Using this and the fact that $(u_{\bar{\lambda}_i}, v_{\bar{\lambda}_i})$ are weak solutions of \eqref{D} it easily follows that there is a strong convergence in $W$ subsequence of $(u_{\bar{\lambda}_i}, v_{\bar{\lambda}_i})$ with limit point $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$. This and \eqref{eq:j-j} yield the required. \end{proof} \section{Asymptotic behaviour of the energy level of ground states of \eqref{D}} In this section we prove statements (I): (b), (c), (II): (b), (c) and (III) of Theorem \ref{thm2}. The proofs will follow from three lemmas below. \begin{lemma}\label{lemm:asympt1} Let $\Omega^+ \neq \emptyset$. Then \begin{itemize} \item $\mathcal{E}(\bar{\lambda}) \to 0$ as $\lambda \uparrow \lambda_1$ and $\mu \uparrow \mu_1$, \item $\mathcal{E}(\bar{\lambda}) \to 0$ as $\bar{\lambda} \to (\lambda_1, \mu_0)$ for any $\mu_0 < \mu_1$, , \item $\mathcal{E}(\bar{\lambda}) \to 0$ as $\bar{\lambda} \to (\lambda_0, \mu_1)$ for any $\lambda_0 < \lambda_1$. \end{itemize} \end{lemma} \begin{proof} All statements are proved in a similar way. We give the proof only for the first statement. Since $n_{\bar{\lambda}} = \mathcal{E}_{\bar{\lambda}}$, it is sufficient to show that $n_{\bar{\lambda}} \to 0$ as $\lambda \to \lambda_1$ and $\mu \to \mu_1$. Consider the first eigenfunction $\varphi_1$. Using the assumption $\Omega^+ \neq \emptyset$ it is not hard to find $w \in W_0^{1,q}$ such that $$Q_\mu(w) > 0 , \quad F(\varphi_1, w) > 0,$$ for all $\mu \in (-\infty, \mu_1]$. This and the fact that $P_\lambda(\varphi_1) > 0$ for $\lambda<\lambda_1$ yield $(\varphi_1, w) \in \mathcal{A}$ and $(t\varphi_1, s w)\in \mathcal{N}_{\bar{\lambda}}$, where $t, s$ are given by \eqref{fiber_t} and \eqref{fiber_s}. At the same time $$P_\lambda(\varphi_1) = (\lambda_1 - \lambda) \int_{\Omega} |\varphi_1|^p \to 0, \quad \text{as } \lambda \to \lambda_1.$$ From here it follows that $$n_{\bar{\lambda}} \leq \mathcal{J}_{\bar{\lambda}}(\varphi_1, w) = C \frac{P_{\lambda}(\varphi_1)^{\alpha/(p d)} Q_{\mu}(w)^{\beta/(q d)}}{F(\varphi_1, w)^{1/d}} \to 0,$$ as $\lambda \to \lambda_1$ and $\mu \to \mu_1$. This completes the proof. \end{proof} The proof of the next lemma can be obtained in the standard way using statement II: (a) of Theorem \ref{thm2}. \begin{lemma} Let $F(\varphi_1, \psi_1) < 0$. Then \begin{itemize} \item $\mathcal{E}(\bar{\lambda}) \to 0$ as $\lambda \downarrow \lambda_1$ and $\mu \downarrow \mu_1$, \item $\mathcal{E}(\bar{\lambda}) \to 0$ as $\bar{\lambda} \to (\lambda_1, \mu_0)$ for any $\mu_0 \in (\mu_1, \mu^*)$ and $\mathcal{E}(\bar{\lambda}) \to 0$ as $\bar{\lambda} \to (\lambda_0, \mu_1)$ for any $\lambda_0 \in (\lambda_1, \lambda^*)$. \end{itemize} \end{lemma} \begin{lemma} If $f(x) \leq 0$, $p, q \geq 2$ and $\max\{p, q\} > 2$, then $\mathcal{E}(\bar{\lambda}) \to -\infty$ as $\bar{\lambda} \to (\lambda^*, \mu^*)$. \end{lemma} \begin{proof} Let $(u^*, v^*)$ be a minimizer of \eqref{mint0} such that $u^*, v^* \geq 0$. Observe that $F(u^*, v^*) = 0$, since by the assumption $f(x) \leq 0$. This implies that $$\label{eq:supp} \Omega \neq \operatorname{supp} u^* \cap \operatorname{supp} v^* \subset \Omega_0 = \{ x \in \Omega: f(x) = 0\}.$$ Assume first that for every minimizer $(u^*, v^*)$ of \eqref{mint0} it holds $$\sigma^* = \frac{1}{\lambda_1}\frac{\int_\Omega |\nabla u^*|^p} {\int_\Omega |u^*|^p} = \frac{1}{\mu_1}\frac{\int_\Omega |\nabla v^*|^q}{\int_\Omega |v^*|^q}.$$ Since $f(x) \leq 0$, by Proposition \ref{sigma>1} we know that $\sigma^* > 1$ and therefore, $\lambda^* > \lambda_1$ and $\mu^* > \mu_1$. This fact, as well as $u^*, v^* \geq 0$ and \eqref{eq:supp} imply that the equations $D_u P_{\lambda^*}(u^*) = 0$ and $D_v Q_{\mu^*}(v^*)= 0$ cannot be satisfied on $\Omega$. Hence there exist $\theta_1, \theta_2 \in C_0^\infty(\Omega)$ such that $$\langle D_u P_{\lambda^*}(u^*), \theta_1 \rangle < 0, \quad \langle D_v Q_{\mu^*}(v^*), \theta_2 \rangle < 0.$$ Consider the functions $$u_r := u^* + r \theta_1, \quad v_r := v^* + r^\delta \theta_2,$$ for $r > 0$ and some constant $\delta>0$ which will be defined below. Then by Taylor's theorem we obtain \begin{gather} \label{eq:pl} P_{\lambda^*}(u_r) = P_{\lambda^*}(u^*) + r \langle D_u P_{\lambda^*}(u^*), \theta_1 \rangle + R_1(r), \\ \label{eq:ql} Q_{\mu^*}(v_r) = Q_{\mu^*}(v^*) + r^\delta \langle D_v Q_{\mu^*}(v^*), \theta_2 \rangle + R_2(r), \end{gather} for sufficiently small $r>0$, where $R_1(r)$ and $R_2(r)$ are reminders, for instance, in the Lagrange form; i.e., \begin{align*} R_1(r) & = r^2 \frac{p (p-1)}{2!} \Big( \int_\Omega |\nabla(u^* + r \kappa \theta_1)|^{p-2} |\nabla \theta_1|^2 dx \\ &\quad - \lambda \int_\Omega |(u^* + r \kappa \theta_1)|^{p-2} |\theta_1|^2 dx \Big), \end{align*} for some $\kappa \in (0,1)$, and similar equality for $R_2(r)$. Since $p, q \geq 2$ by assumption, we guarantee that $R_1(r) = o(r)$ and $R_2(r) = o(r^\delta)$. Using this fact and the observation that $P_{\lambda^*}(u^*) = 0$, $Q_{\mu^*}(v^*) = 0$, from \eqref{eq:pl} and \eqref{eq:ql} we obtain \begin{gather} \label{eq:p<<0} P_{\lambda^*}(u_r) = r \langle D_u P_{\lambda^*}(u^*), \theta_1 \rangle + o(r) < 0, \\ \label{eq:q<<0} Q_{\mu^*}(v_r) = r^\delta \langle D_v Q_{\mu^*}(v^*), \theta_2 \rangle + o(r^\delta) < 0, \end{gather} for sufficiently small $r>0$. On the other hand, we have \begin{align*} F(u_r, v_r) &= r^{ \delta \beta} F_1 + r^\alpha F_2 \\ &:= r^{\delta \beta} \int_{\Omega_{u^*} \setminus \Omega_0} f |u^* + r \theta_1|^\alpha |\theta_2|^\beta \, dx +r^\alpha \int_{\Omega_{v^*} \setminus \Omega_0} f |\theta_1|^\alpha |v^* + r^\delta \theta_2|^\beta \, dx \\ &< 0. \end{align*} Here it holds a strong inequality. Indeed, if one suppose $F_1 = 0$ and $F_2 = 0$, then $(u_r, v_r)$ would be an admissible point for \eqref{mint0} but by \eqref{eq:p<<0}, \eqref{eq:q<<0} $P_{\lambda^*}(u_r) < 0$, $Q_{\mu^*}(v_r) < 0$ that contradicts the definition of $\sigma^*$. Let $t_r$ and $s_r$ are given by \eqref{fiber_t} and \eqref{fiber_s}. Then $(t_r u_r, s_r v_r) \in \mathcal{N}_{(\lambda^*, \mu^*)}$ and \label{eq:j*} \begin{aligned} &\mathcal{J}_{({\lambda}^*, \mu^*)}(u_r, v_r)\\ & = - C \frac{|P_{\lambda^*}(u_r)|^{\alpha/(p d)} |Q_{\mu^*}(v_r)| ^{\beta/(q d)}}{|F(u_r, v_r)|^{1/d}} \\ &= - C \frac{r^{\frac{\alpha}{p d} + \frac{\delta \beta}{q d}} |\langle D_u P_{\lambda^*}(u^*), \theta_1 \rangle+o(r) /r|^{\alpha/(p d)} |\langle D_v Q_{\mu^*}(v^*), \theta_2 \rangle+o(r^\delta)/r^\delta|^{\beta/(q d)}}{|r^{\delta \beta} F_1 + r^{\alpha} F_2|^{1/d}}. \end{aligned} We will prove the theorem, if we find $\delta > 0$, for which the system \begin{gather*} \delta \beta - \frac{\alpha}{p} - \frac{\delta \beta}{q} > 0, \\ \alpha - \frac{\alpha}{p} - \frac{\delta \beta}{q} > 0, \end{gather*} will be consistent. Expressing $\delta$ from the first and second inequalities, we obtain $$\frac{\alpha q}{\beta p} \frac{1}{(q - 1)} < \delta < \frac{\alpha q}{\beta p} (p-1).$$ From the assumptions $p, q \geq 2$ and $\max \{p, q\} > 2$ we conclude that such $\delta$ exists and therefore, $$\label{eq:jtoinf} E_{({\lambda}^*, \mu^*)}(t_r u_r, s_r v_r) =\mathcal{J}_{({\lambda}^*, \mu^*)}(u_r, v_r) \to -\infty \quad \text{as } r \to 0,$$ Hence $n_{({\lambda}^*, \mu^*)} = -\infty$. Assume now that there exists a minimizer $(u^*, v^*)$ of \eqref{mint0} such that $$\sigma^* = \frac{1}{\lambda_1}\frac{\int_\Omega |\nabla u^*|^p} {\int_\Omega |u^*|^p} > \frac{1}{\mu_1}\frac{\int_\Omega |\nabla v^*|^q}{\int_\Omega |v^*|^q}.$$ In this case the proof is actually the same as in the previous case except that now we have to take into account that $Q_{\mu^*}(v^*) < 0$ and therefore \eqref{eq:j*} will be changed. Let us indicate the proof briefly. Since $\lambda^* > \lambda_1$, we can find $\theta$ such that $\langle D_u P_{\lambda^*}(u^*), \theta \rangle < 0$. Consider the function $u_r := u^* + r \theta$, for $r > 0$. Then, in view of assumption $p \geq 2$, we have $$P_{\lambda^*}(u_r) = P_{\lambda^*}(u^*) + r \langle D_u P_{\lambda^*}(u^*), \theta \rangle + o(r) = r \langle D_u P_{\lambda^*}(u^*), \theta \rangle + o(r) < 0,$$ for sufficiently small $r>0$. Furthermore, we have $$F(u_r, v^*) = r^\alpha F_3:= r^\alpha \int_{\Omega \setminus \Omega_0} f |\theta|^\alpha |v^*|^\beta \, dx < 0.$$ Thus, we obtain \begin{align*} \mathcal{J}_{(\lambda^*, \mu^*)}(u_r, v^*) & = \frac{|Q_{\lambda^*}(v^*)|^{\beta/(q d)} |P_{\lambda^*}(u_r)|^{\alpha/(p d)}}{|F(u_r, v^*)|^{1/d}} \\ &= \frac{r^{\alpha/(p d)} |\langle D_u P_{\lambda^*}(u^*), \theta \rangle+o(r)/r|^{\alpha/(p d)} |Q_{\lambda^*}(v^*)|^{\beta/(q d)}} {r^{\alpha/d} |F_3|^{1/d}}. \end{align*} Hence we again get \eqref{eq:jtoinf}. Note that in this case we have not used the assumption $\max\{\alpha, \beta\} > 2$. To conclude the proof, we observe that for any fixed $(u,v) \in W$ the function $\mathcal{J}_{\bar{\lambda}}(u, v)$ is a continuous map with respect to $\bar{\lambda}$. Furthermore $\mathcal{E}_{\bar{\lambda}} \leq \mathcal{J}_{\bar{\lambda}}(u_r, v_r)$ for any $\bar{\lambda} \in \Sigma^*$. These and \eqref{eq:jtoinf} imply that $\mathcal{E}_{\bar{\lambda}} \to -\infty$ as $\lambda \to \lambda^*$, $\mu \to \mu^*$. \end{proof} \section{Blow-up results} In this Section we prove Theorems \ref{thm3} and \ref{thm4}. \begin{proof}[Proof of Theorem \ref{thm3}] The proofs of statements (1) and (2) are similar; so we give the proofs of (1) and (3) only. \noindent{\it Proof of statement (1)}. Let $(u_{\bar{\lambda}}, v_{\bar{\lambda}})$ be a solution of \eqref{D} and $\bar{\lambda} \to (\lambda_1, \mu_0)$, where $\mu_0 < \mu_1$. Let $$u_{\bar{\lambda}} = t_{\bar{\lambda}} \hat{u}_{\bar{\lambda}}, \quad v_{\bar{\lambda}} = s_{\bar{\lambda}} \hat{v}_{\bar{\lambda}},$$ where $\|\hat{u}_{\bar{\lambda}}\|_q = 1$, $\|\hat{v}_{\bar{\lambda}}\|_q = 1$ and $t_{\bar{\lambda}}=\|u_{\bar{\lambda}}\|_q$, $s_{\bar{\lambda}}=\|v_{\bar{\lambda}}\|_q$. From \eqref{fiber_J} and Lemma \ref{lemm:asympt1} we know that $$\label{eq:j=jto0} \mathcal{J}_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) = \mathcal{J}_{\bar{\lambda}}(\hat{u}_{\bar{\lambda}}, \hat{v}_{\bar{\lambda}}) = C \frac{P_{\lambda}(\hat{u}_{\bar{\lambda}})^{\alpha/(p d)} Q_{\mu}(\hat{v}_{\bar{\lambda}})^{\beta/(q d)}}{F(\hat{u}_{\bar{\lambda}}, \hat{v}_{\bar{\lambda}})^{1/d}} \to 0,$$ as $\bar{\lambda} \to (\lambda_1, \mu_0)$. At the same time, since $\mu_0 < \mu_1$, we have $$Q_\mu(\hat{v}_{\bar{\lambda}}) = 1 - \mu \int_\Omega |\hat{v}_{\bar{\lambda}}|^q \, dx > 1 - \frac{\mu}{\mu_1} > 0$$ uniformly by $\bar{\lambda}$. Therefore from \eqref{eq:j=jto0} it follows that $$\label{eq:p/fto0} \frac{P_{\lambda}(\hat{u}_{\bar{\lambda}})^{\alpha/(p d)}} {F(\hat{u}_{\bar{\lambda}}, \hat{v}_{\bar{\lambda}})^{1/d}} = \Big( \frac{P_{\lambda}(\hat{u}_{\bar{\lambda}})^{\alpha}} {F(\hat{u}_{\bar{\lambda}}, \hat{v}_{\bar{\lambda}})^{p}} \Big)^{1/(pd)} \to 0.$$ Hence, from \eqref{fiber_s} we obtain that $s_{\bar{\lambda}} \to 0$ and consequently $\|v_{\bar{\lambda}}\|_q \to 0$ as $\bar{\lambda} \to (\lambda_1, \mu_0)$. From \eqref{eq:p/fto0} it follows that $P_{\lambda}(\hat{u}_{\bar{\lambda}}) \to 0$, since $F(\hat{u}_{\bar{\lambda}}, \hat{v}_{\bar{\lambda}})$ is bounded. This, \eqref{fiber_t} and the assumption $q < \beta$ imply that $t_{\bar{\lambda}} \to +\infty$ and consequently $\|u_{\bar{\lambda}}\|_p \to +\infty$ as $\bar{\lambda} \to (\lambda_1, \mu_0)$. \smallskip \noindent\emph{Proof of statement (3)}. Suppose $(\bar{\lambda}_m)$, $m \in \mathbb{N}$ is a sequence in $\Sigma_1$ such that $\lambda_m \to \lambda_1$ and $\mu_m \to \mu_1$ as $m\to \infty$. As above, consider $u_{\bar{\lambda}_m} = t_{\bar{\lambda}_m} \hat{u}_{\bar{\lambda}_m}$ and $v_{\bar{\lambda}_m} = s_{\bar{\lambda}_m} \hat{v}_{\bar{\lambda}_m}$, $m=1,2,\dots$. Since $\|\hat{u}_{\bar{\lambda}_m}\|_p=1$ and $\|\hat{v}_{\bar{\lambda}_m}\|_q=1$ for $m \in \mathbb{N}$, we can apply the Eberlein-Shmulyan theorem and the Sobolev embedding theorem. Thus, there exist $(\hat{u}_0,\hat{v}_0) \in W$ and a subsequence, which we denote again $(\bar{\lambda}_m)$, such that $(\hat{u}_{\bar{\lambda}_m},\hat{v}_{\bar{\lambda}_m}) \to (\hat{u}_0,\hat{v}_0)$ as $m\to \infty$ weakly in $W$ and strongly in $L_r(\Omega)\times L_s(\Omega)$ as $r \in (1,p^*)$, $s \in (1,q^*)$. For $(u_{\bar{\lambda}_m}, v_{\bar{\lambda}_m}) \in \mathcal{N}_{\bar{\lambda}_m}$ we have $$\label{sh1} E_{\bar{\lambda}_m}(u_{\bar{\lambda}_m}, v_{\bar{\lambda}_m}) = t_{\bar{\lambda}_m}^p\frac{d }{\alpha} P_{\lambda_m}(\hat{u}_{\bar{\lambda}_m}) =s_{\bar{\lambda}_m}^q\frac{d}{\beta} Q_{\lambda_m}(\hat{v}_{\bar{\lambda}_m}) \to 0\quad\text{as } m\to \infty.$$ Suppose, for instance, that $\hat{v}_0= 0$. Then $Q_{\mu}(\hat{v}_{\bar{\lambda}_m}) \to 1$. This allows us to apply the arguments from the proof of statement (1). Indeed, \eqref{sh1} implies that $s_{\bar{\lambda}_m} \to 0$ as $m\to \infty$. Furthermore, similar to \eqref{eq:p/fto0} it is deduced that $$\frac{P_{\lambda_m}(\hat{u}_{\bar{\lambda}_m})^{\alpha}} {F(\hat{u}_{\bar{\lambda}_m}, \hat{v}_{\bar{\lambda}_m})^{p}} \to 0.$$ Then $P_{\lambda_m}(\hat{u}_{\bar{\lambda}_m}) \to 0$ and due to \eqref{fiber_t} and the assumption $q < \beta$ we have $t_{\bar{\lambda}} \to +\infty$. Thus in this case statement (3) is true. Consider now the case $\hat{u}_0\neq 0$ and $\hat{v}_0\neq 0$. Suppose that simultaneously $P_{\lambda_m}(\hat{u}_{\bar{\lambda}_m})\to 0$ and $Q_{\lambda_m}(\hat{v}_{\bar{\lambda}_m}) \to 0$ as $m\to \infty$. Then by the weak lower semicontinuity of $P_\lambda(u)$, $Q_\lambda(v)$ it follows that $P_{\lambda_1}(\hat{u}_0) \leq 0$ and $Q_{\mu_1}(\hat{v}_0) \leq 0$. This is possible only if $\hat{u}_0=\varphi_1$ and $\hat{v}_0=\psi_1$ up to nonzero multiplier. Hence we have $0\leq F(\hat{u}_0, \hat{v}_0)=F(\varphi_1,\psi_1)$. However this contradicts the assumption $F(\varphi_1,\psi_1)<0$. Thus we can assume without loss of generality that $Q_{\lambda_m}(\hat{v}_{\bar{\lambda}_m}) \to c_1>0$ as $m\to \infty$. However we are again in the position to apply the arguments from the proof of statement (1) and therefore we obtain anew the desired $s_{\bar{\lambda}_m} \to 0$, $t_{\bar{\lambda}_m} \to +\infty$ as $m\to \infty$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm4}] Note that for $(u_{\bar{\lambda}}, v_{\bar{\lambda}}) \in \mathcal{N}_{\bar{\lambda}}$ we have $$E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) = \big( \frac{\alpha}{p} + \frac{\beta}{q} - 1 \big) \frac{1}{\alpha} P_\lambda(u_{\bar{\lambda}}).$$ From here it follows that $$\label{eq:e-p} E_{\bar{\lambda}}(u_{\bar{\lambda}}, v_{\bar{\lambda}}) \to -\infty \Longleftrightarrow P_\lambda(u_{\bar{\lambda}}) \to -\infty,$$ as $\bar{\lambda} \to (\lambda^*, \mu^*)$. On the other hand $$P_\lambda(u_{\bar{\lambda}}) = \|u_{\bar{\lambda}}\|_p^p - \lambda \int_\Omega |u_{\bar{\lambda}}|^p \, dx \geq -\frac{\lambda}{\lambda_1} \|u_{\bar{\lambda}}\|_p^p.$$ From here and \eqref{eq:e-p} we obtain $\|u_{\bar{\lambda}}\|_p \to +\infty$. By the same arguments it also follows that $\|v_{\bar{\lambda}}\|_q \to +\infty$. \end{proof} \section{Final remarks} Let us compare the obtained results for \eqref{D} in the special case $p=q$, $\lambda = \mu$ with known results for \eqref{pb} when $\gamma = \alpha + \beta$. In \cite{ilst, ilIZV} to study \eqref{pb} the following critical value is introduced $$\lambda^*_{\rm sing} = \inf_u \big\{\frac{\int |\nabla u|^p \, dx}{\int |u|^p \, dx}: F(u):=\int f|u|^\gamma \, dx \geq 0,\; u \in W^1_p(\Omega)\big\},$$ and under the same assumptions of Corollary \ref{cor1} the following is proven: \begin{itemize} \item[(I)] $\lambda^*_{sing}<+\infty$ if and only if $\Omega^{0}\cup \Omega^+ \neq \emptyset$; \item[(II)]$\lambda_1<\lambda^*_{sing}$ if and only if $F(\phi_1) < 0$; \item[(III)] problem \eqref{pb} has two sets of positive solutions: \begin{itemize} \item[(1)] $w_\lambda^1$ for $\lambda\in (-\infty,\lambda^*_{sing})$ in the case $\Omega^+\neq \emptyset$, such that $E_\lambda(w_\lambda^1) > 0$; \item[(2)] $w_\lambda^2$ for $\lambda \in (\lambda_1,\lambda^*_{sing})$ in the case $F(\phi_1) < 0$, such that $E_\lambda(w_\lambda^2) < 0$; \end{itemize} \item[(IV)] the solutions $w_\lambda^1$ on $(-\infty,\lambda_1)$ and $w_\lambda^2$ on $(\lambda_1,\lambda^*_{sing})$ are the ground states of \eqref{pb} and the functions $\mathcal{E}^1(\lambda):=E_\lambda(w_\lambda^1)$, $\mathcal{E}^2(\lambda):=E_\lambda(w_\lambda^2)$ are continuous in the intervals of their determination. \end{itemize} To compare \eqref{pb} and \eqref{D} it is necessary also to take into consideration that the critical values $\lambda^*$ and $\lambda^*_{sing}$ are essentially the same objects but for different problems. For instance, both of them define a threshold of the applicability of Nehari manifold method (see Section 4, and \cite{ilyas, ilst, ilIZV}). Moreover, they can be obtained as a consequence of a general approach (see \cite{ilst, ilIZV}). First, we see that in contrast to \eqref{pb} we always have $\lambda^*<+\infty$ for \eqref{D}. The next distinction is that in the case $F(\phi_1)< 0$, $\Omega^+\neq \emptyset$ the ground state level of \eqref{pb} has a discontinuity at the point $\lambda_1$. Furthermore, in the interval $(\lambda_1,\lambda^*_{sing})$ one has a multiplicity of solutions to \eqref{pb}; i.e., there are two branches positive solutions $w_\lambda^1$ and $w_\lambda^2$, whereas this property is not observed for \eqref{D} (see Figure \ref{fig2}). \begin{figure}[ht] \begin{center} \includegraphics[width=0.95\textwidth]{fig2} \end{center} \caption{$p=q$, $\lambda=\mu$, $F(\varphi_1, \varphi_1)<0$, $\Omega_+\neq\emptyset$} \label{fig2} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[width=0.95\textwidth]{fig3} \end{center} \caption{$p=q$, $p>2$, $\lambda=\mu$, $f(x)<0$} \label{fig3} \end{figure} For \eqref{pb} similar results like in Theorem \ref{thm2} are known \cite{ilIZV}. In particular, $\mathcal{E}^1(\lambda) \to 0$ as $\lambda \uparrow \lambda^*_{sing}$ and $\mathcal{E}^2(\lambda) \to 0$ as $\lambda \downarrow \lambda_1$. Additionally, $\mathcal{E}^2(\lambda) \to -\infty$ as $\lambda \to \lambda^*_{sing}$ if and only if $\Omega^+ = \emptyset$, where $\lambda^*_{sing}<+\infty$ if and only if $\Omega^0 \neq \emptyset$. Thus, we see that in case $f(x) < 0$ and $p > 2$ ground states of the system \eqref{D} blow up at the finite value $\lambda^*$, whereas this phenomenon for the scalar equation \eqref{pb} is impossible (see Figure \ref{fig3}). \begin{remark} \label{rmk10.1} \rm Consider the positive solutions $w^1_\lambda$ and $w^2_\lambda$ of \eqref{pb} for $\lambda \in (-\infty, \lambda^*)$ and $\lambda \in (\lambda_1,\lambda^*)$, respectively. Then it easy to verify that the following pairs of functions $u_\lambda^1=c_1 w^1_\lambda$, $v_\lambda^1=c_2 w^1_\lambda$ and $u_\lambda^2=c_1 w^2_\lambda$, $v_\lambda^2=c_2 w^2_\lambda$, where $$c_1=\frac{\alpha^{(\beta-p)/(p^2d)}}{\beta^{\beta/(p^2d)}},\quad c_2=\frac{\beta^{(\alpha-p)/(p^2d)}}{\alpha^{\alpha/(p^2d)}},$$ satisfy the system \eqref{D} with $p=q$, $\lambda=\mu$, $\alpha+\beta=\gamma$ in the intervals $(-\infty, \lambda^*)$ and $(\lambda_1,\lambda^*)$, respectively. It is interesting to compare these solutions with those obtained in Corollary \ref{cor1}. The properties of the corresponding energy functionals $\mathcal{E}_\lambda$ (see Corollary \ref{cor2} and Figures \ref{fig2} and \ref{fig3}) show that these solutions are different. The differences can be seen also from the blow-up behaviour of the ground state branches of \eqref{D} obtained in Theorem \ref{thm3},\ref{thm4}, which, as it is easy to see, is impossible for the ground state branches of \eqref{pb} under the same assumptions. \end{remark} \begin{remark} \label{rmk10.2} \rm Observe that $n_{\bar{\lambda}} = 0$ when $\bar{\lambda}$ belongs to quadrant II or III (see Figure \ref{fig1}). This can be seen from the following. First of all, note that the set $\mathcal{B}$ is empty when $\bar{\lambda}$ lies in quadrants II and III. Further, if we consider, for example, quadrant III; i.e., $\lambda > \lambda_1$, $\mu < \mu_1$, then one can find a sequence of functions $(\xi_k, \zeta_k) \in W$ such that $P_\lambda(\xi_k) > 0$, $Q_\mu(\zeta_k) > 0$, $F(\xi_k, \zeta_k) > c_0 > 0$ uniformly by $k \in \mathbb{N}$ and $P_\lambda(\xi_k) \to 0$. \end{remark} \begin{remark} \label{rmk10.3} \rm As it was noted above, the scalar problem \eqref{pb} in case $F(\phi_1)< 0$ has multiple positive solutions $w_\lambda^1$, $w_\lambda^2$ for $\lambda \in (\lambda_1,\lambda^*_{sing})$. On the other hand it is known (see \cite{ilfunc, Ou2}) that this problem possesses a turning point $(\lambda_0,w_0)$, where $\lambda_0>\lambda_{sing}^*$ and $w_0$ is a positive solution of \eqref{pb}. Thus we can assume up to the uniqueness of positive branches $(w_\lambda^1)$, $(w_\lambda^2)$ that they are linked at the point $(\lambda_0,w_0)$. However, we did not find for \eqref{D} the second branch of solutions which could be considered as a contender to link with the branch of ground states $(u_\lambda, v_\lambda)$ found in Theorems \ref{thm1}, \ref{thm2}. In view of this, we conjecture that there are only two scenarios of the behaviour of the ground state branch of \eqref{D}: it continues on the whole quadrant IV or there is a threshold in IV where it blows up. \end{remark} \begin{remark} \label{rmk10.4} \rm It is known that for the scalar problem \eqref{pb}, the assumption $F(\phi_1)< 0$ is necessary and sufficient for the existence of positive solutions for $\lambda > \lambda_1$. We cannot prove the similar statement for \eqref{D}. In particular it is an open question whether \eqref{D} has positive solutions outside of the quadrant I, if $F(\varphi_1, \psi_1) \geq 0$. \end{remark} \begin{remark} \label{rmk10.5} \rm In this paper we consider only the case $d = \frac{\alpha}{p} + \frac{\beta}{q} - 1 > 0$ which corresponds to the case when Hessian $\Gamma_{\bar{\lambda}}(u, v)$ is indefinite. The cases $d=0$ and $d<0$ for \eqref{D} have been investigated in the literature (see e.g. \cite{Bockard,Felmer,Figueiro, Figueiro2}). However, in these cases, to our knowledge, there are no investigations similar to those obtained in the present paper. \end{remark} \subsection*{Acknowledgments} The authors would like to express their gratitude to Professor Dumitru Motreanu, for helpful discussions and encouragement. The first author was partly supported by grant RFBR 13-01-00294-p-a. The second author was partly supported by grant RFBR 11-01-00348-p-a \begin{thebibliography}{00} \bibitem{Alam} S. Alama, G. Tarantello; \emph{On semilinear elliptic problems with indefinite nonlinearities}. Calculus Var. and Partial Differential Equations 1, 439-475, (1993) \bibitem{Alleg} W. Allegretto, Y. X. Huang; \emph{A Picone's identity for the $p$-Laplacian and applications}. Nonlinear Anal., Theory Methods Appl. 32, No.7, 819-830, (1998) \bibitem{Alves} C. O. Alves, D. C. de Morais Filho, M. A. S. Souto; \emph{On systems of elliptic equations involving subcritical or critical Sobolev exponents}. Nonlinear Anal., Theory Methods Appl. 42, No. 5, A, 771-787, (2000) \bibitem{Anan} A. Anane; \emph{Etude des valeurs propres et de la r\'esonance pour l'op\'erateur $p$-Laplacien}, Th\`ese de doctorat, Universit\'e Libre de Bruxelles, Brussels, (1988) \bibitem{Bartsch} T. Bartsch, Y. Guo; \emph{Existence and nonexistence results for critical growth polyharmonic elliptic systems}, Journal of Differential Equations V 220, Issue 2, 531–-543, (2006) \bibitem{Bandle} C. Bandle, M. A. Pozio, A. Tesei; \emph{Existence and uniqueness of solutions of nonlinear Neumann problems}. Math. Z. 199, No.2, 257-278, (1988) \bibitem{BCDN} H. Berestycki, I. Capuzzo-Dolcetta, L. Nirenberg; \emph{Superlinear indefinite elliptic problems and nonlinear Liouville theorem}, Topological Methods in Nonlinear Analysis 4, 59-78, (1994) \bibitem{BCDN1} H. Berestycki, I. Capuzzo-Dolcetta, L. Nirenberg; \emph{Variational methods for indefinite superlinear homogeneous elliptic problems}, NoDEA 2, 553-572, (1995) \bibitem{Bockard} L. Boccardo, D. G. de Figueiredo; \emph{Some remarks on a system of quasilinear elliptic equations}, NoDEA 9, pp 309-323, (2002) \bibitem{Bozhkov} Y. Bozhkov, E. Mitidieri; \emph{Existence of multiple solutions for quasilinear systems via fibering method}, J. Differ. Equations 190, No.1, 239-267, (2003) \bibitem{Clement} P. H. Clement, D. G. de Figueiredo, E. Mitidieri; \emph{Positive solutions of semilinear elliptic systems}, Comm. Partial Differential Equations 17, (1992) \bibitem{Col} S. Coleman, V. Glazer, A. Martin; \emph{Action minima among solution to a class of Euclidean scalar field equations}, Comm. Math. Phys. 58, 2, 211-221, (1978) \bibitem{Costa} D. G. Costa; \emph{On a Class of Elliptic Systems in $\mathbb{R}^N$}, Electron. J. Diff. Eqns. Vol 1994, No. 7, 1-14, (1994) \bibitem{delpin} M. Del Pino; \emph{Positive solutions of a semilinear elliptic equation on a compact manifold}, Nonlinear Anal., 22, no. 11, 1423-1430, (1994) \bibitem{delpin2} M. Del Pino, P. L. Felmer; \emph{Multiple solutions for a semilinear elliptic equation}, Trans. Amer. Math. Soc., 347, no. 12, 4839-4853, (1995) \bibitem{DrabekT} P. Drabek, P. Krejci, P. Takac; \emph{Nonlinear differential equations}. Chapman \& Hall/CRC Boca Raton, FL, (1999) \bibitem{drabek} P. Drabek, S. I. Pohozaev; \emph{Positive solution for the p-Laplacian: application of the fibering method}, Proc. Roy. Soc. Edinb. Sect. A, 127, 703-726, (1997) \bibitem{Felmer} P. Felmer, R. F. Manasevich, F. de Thelin; \emph{Existence and Uniqueness of Positive Solutions for Certain Quasilinear Elliptic Systems}, Comm. Partial Differential Equations Volume 17, Issue 11-12, (1992) \bibitem{Figueiro} D. de Figueiredo; \emph{Semilinear elliptic systems a survey of superlinear problems}, Resenhas IME USP 2, 373-391, (1996) \bibitem{Figueiro2} D. de Figueiredo; \emph{Semilinear elliptic systems: Existence, multiplicity, symmetry of solutions}, Handbook of Differential Equations: Stationary Partial Differential Equations, Volume 5, Pages 1-48, (2008) \bibitem{ilyas} Y. Il'yasov; \emph{Euler's functional for equations involving the $p$-Laplacian as a functional of the spectral parameter}, Proc. Stekl. Ins. Math., 214, 175-186, (1996) \bibitem{ilcomp} Y. Il'yasov; \emph{On positive solutions of indefinite elliptic equations}, C. R. Acad. Sci., Paris 332, S\'{e}rie I, 1-6, (2001) \bibitem{ilst} Y. Il'yasov; \emph{On a procedure of projective fibering of functionals on Banach spaces}, Proc. Steklov Inst. Math., 232, 150-156, (2001) \bibitem{ilIZV} Y. Il'yasov; \emph{Non-local investigation of bifurcations of solutions of non-linear elliptic equations}, Izv. RAN. Ser. Mat., 66:6, (2002) \bibitem{ilconcan} Y. Il'yasov; \emph{On nonlocal existence results for elliptic equations with convex-concave nonlinearities}, Nonlinear Analysis: Theory, Methods \& Applications, 61 (1-2), 211-236, (2005) \bibitem{ilrunst} Y. Il'yasov, T. Runst; \emph{Positive solutions for indefinite inhomogeneous Neumann elliptic problems}, Electronic Journal of Differential Equations, 57, 1-21, (2003) \bibitem{ilfunc} Y. Il'yasov; \emph{Bifurcation Calculus by the Extended Functional Method}, Functional Analysis and Its Applications, 41:1, 18–-30, (2007) \bibitem{ilegorov} Y. Il'yasov, Y. Egorov; \emph{Hopf boundary maximum principle violation for semilinear elliptic equations}, Nonlinear Analysis: Theory, Methods \& Applications 72 (7-8), 3346-3355, (2010) \bibitem{Neh} Z. Nehari; \emph{Characteristic values associated with a class of nonlinear second-order differential equations}, Acta Math. 105, 141-175, (1961) \bibitem{lindqvist} P. Lindqvist; \emph{On the equation} $\operatorname{div}(| \nabla u| \sp{p-2}\nabla u) +\lambda | u| \sp{p-2}u=0$, Proc. Am. Math. Soc. 109, No.1, 157-164, (1990) \bibitem{Ou1} T. C. Ouyang; \emph{On the positive solutions of semilinear equations $\Delta u+\lambda u+hu^p=0$ on the compact manifolds}, Indiana Univ. Math. J. 40, 1083-1141, (1991) \bibitem{Ou2} T. C. Ouyang; \emph{On the positive solutions of semilinear equations $\Delta u+\lambda u+hu^p=0$ on the compact manifolds, Part II}, Indiana Univ. Math. J. 40, 1083-1141, (1991) \bibitem{Poh} S. I. Pohozaev; \emph{On an approach to Nonlinear equations}. Doklady Acad. Sci. USSR, 247, 1327-1331, (1979) \bibitem{Poh2} S. I. Pohozaev; \emph{On the method of fibering a solution in nonlinear boundary value problems}, Proc. Stekl. Ins. Math., 192, 157-173, (1990) \bibitem{Szulkin} A. Szulkin, T. Weth; \emph{The method of Nehari manifold}. D. Y. Gao (ed.) et al., Handbook of nonconvex analysis and applications. Somerville, MA: International Press. 597-632, (2010) \end{thebibliography} \end{document}