\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Sixth Mississippi State Conference on Differential Equations and Computational Simulations, {\em Electronic Journal of Differential Equations}, Conference 15 (2007), pp. 107--126.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{107} \title[\hfilneg EJDE-2006/Conf/15\hfil Positive solutions for elliptic problems] {Positive solutions for elliptic problems with critical indefinite nonlinearity in bounded domains} \author[J. Giacomoni, J. Prajapat, M. Ramaswamy\hfil EJDE/Conf/15 \hfilneg] {Jacques Giacomoni, Jyotshana V. Prajapat, Mythily Ramaswamy} % in alphabetical order \address{Jacques Giacomoni \newline MIP-CEREMATH/bat C, Manufacture des Tabacs\\ All\'ee de Brienne 21\\ 31000 Toulouse, France} \email{giacomo4@yahoo.fr} \address{Jyotshana V. Prajapat \newline School of Mathematics\\ Tata Institute of Fundamental Research\\ Homi Bhabha Road, Mumbai 400 005, India} \email{jyotsna@math.tifr.res.in} \address{Mythily Ramaswamy \newline TIFR Center, IISc. Campus \\ Bangalore 560 012, India} \email{mythily@math.tifrbng.res.in} \thanks{Published February 28, 2007.} \subjclass[2000]{35J60, 35B45, 35B33, 35B32} \keywords{Critical indefinite nonlinearity; bifurcation; a priori estimates} \begin{abstract} In this paper, we study the semilinear elliptic problem with critical nonlinearity and an indefinite weight function, namely $$ - \Delta u =\lambda u + h (x) u^{(n+2)/(n-2)} $$ in a smooth open bounded domain $\Omega\subseteq \mathbb{R}^n$, $n > 4 $ with Dirichlet boundary conditions and for $\lambda \geq 0 $. Under suitable assumptions on the weight function, we obtain the positive solution branch, bifurcating from the first eigenvalue $\lambda_1(\Omega)$. For $n=2$, we get similar results for $-\Delta u =\lambda u + h (x)\phi(u)e^u$ where $\phi$ is bounded and superlinear near zero. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} In this paper, we study the following (critical exponent) semilinear elliptic problem in an open bounded domain $\Omega\subseteq\mathbb{R}^n$ with smooth boundary \begin{equation}\label{Eq1.1} \begin{gathered} - \Delta u = \lambda u + h (x) u^{\frac{n+2}{n-2}}\quad \text{in } \Omega \\ u > 0 \quad \text{in } \Omega; \quad u = 0 \quad \text{on } \partial\Omega \end{gathered} \end{equation} for dimensions $n > 4$, $\lambda$ a nonnegative parameter and $h $ a $C^2$ function which changes sign. If $n=2$, we are interested in the following corresponding critical problem \begin{equation}\label{Eq1.2} \begin{gathered} - \Delta u = \lambda u + h (x) \phi(u)e^u\quad \text{in } \Omega \\ u > 0 \quad \text{in } \Omega; \quad u = 0 \quad \text{on } \partial\Omega . \end{gathered} \end{equation} Concerning $h$, we assume the following hypotheses: \begin{itemize} \item[(H1)] $h$ belongs to $C^2(\bar{\Omega})$, \item[(H2)] $h$ could change sign: Denoting by $\Omega^+:=\{x\in \Omega: h(x)>0\}$ and by\\ $\Omega^-:=\{x\in \Omega: h(x)<0\}$, we have $\Omega^+\neq\emptyset$ and $\partial\Omega\subset (\{h>0\}\cup\{h<0\})$. \item[(H3)] $\Gamma:={\overline {\Omega^+}}\cap {\overline{\Omega^-}} \subset \Omega$ with $\nabla h(x)\neq 0$ for all $x\in\Gamma$. \item[(H4)] $\Omega^0:=\Omega\backslash\overline{\Omega^+\cup \Omega^-} $, possibly empty, satisfies: \begin{enumerate} \item $\overline{\Omega^0}\subset\Omega^-$, \item $\partial\Omega^0\cap\partial\Omega^+=\emptyset$, and \item $\lambda_1(\Omega^0)>\lambda_1(\Omega^+)$, where $\lambda_1(\cdot)$ is the first eigenvalue of $-\Delta$ in $\cdot$ with Dirichlet boundary conditions. \end{enumerate} \end{itemize} Notice that either $\partial\Omega \subset \{h>0\}$ or $\partial\Omega \subset \{h<0\}$ since $\Gamma \cap \partial\Omega $ is empty. We assume near each point $\bar{x}\in S_1=\{x\in\Omega^+|\nabla h(x)=0\}$, either one of the following flatness conditions holds: \begin{itemize} \item[(H5a)] $ h(x)=h(\bar{x})+\sum_{i=1}^na_i|x_i-\bar{x}_i|^{\beta-1}(x-\bar{x})_i\,+R(x)$ with $ \nabla R(x)=o(|x-\bar{x}|^{\beta})$, $a_i\neq 0$ , for all $i$ and $ n-2 <\beta< n$ if $n \geq 5$. \item[(H5b)] $c \mathop{\rm dist}(x, S_1)^{\beta-1}\leq |\nabla h(x)|\leq C\mathop{\rm dist}(x,S_1)^{\beta-1}$ for all $x\in\Omega^+$ with $c,C>0$ and, $c_1|x_i|^{\beta}\leq x \cdot \nabla h(x+\bar{x})$ for $x\in B_{\sigma_0}(\bar{x})$ with $i\in (1,\dots ,n)$, $c_1, \sigma_0>0$ and \begin{gather*} n-2 < \beta < n \quad\text{for } n \geq 6\\ n-2 < \beta < n-1 \quad\text{for } n = 5. \end{gather*} \end{itemize} As $h$ is $C^2$, using Taylor's expansion, $$ x \cdot \nabla h(x+\bar{x}) = \langle 2 D^2 h(\bar{x}) , x \rangle + o (|x|^2). $$ As $ \beta > n-2 \geq 2$, this condition is likely to hold at minimum points of $h$ but not at maximum points. Concerning $\phi$, we assume, as in \cite{AdGi}, that \begin{itemize} \item[(H6)]$\phi$ is bounded, $C^1(\mathbb{R},\mathbb{R}^+)$ and $C_1 u^{p'}\leq\phi(u)\leq Cu^{p}$ in a neighborhood of $0$, with $p'\geq p >1$ and $C_1,C>0$. Moreover, $\phi(u)>C_0>0$, for $\|u\|$ large. \item [(H7)] $\phi '$ is bounded and such that $\phi +\phi '\geq 0$. \end{itemize} Under the above assumptions, we have the following result. \begin{theorem}\label{th1} \begin{enumerate} \item Suppose that assumptions (H1)-(H4) and (H5a) or (H5b) are satisfied. Then, there exists a continuum of positive solutions, $\mathcal{C}$, to \eqref{Eq1.1} in $\mathbb{R}^+\times C^1_0(\Omega)$ bifurcating from $\lambda_1(\Omega)$ and satisfying \begin{itemize} \item[(i)] $\Pi_{\mathbb{R}}\mathcal{C}=[0,\lambda_*]$ where $\lambda_1(\Omega)\leq\lambda_*<\lambda_1(\Omega^+)$. \item[(ii)] If $\int_{\Omega}h{\phi^1_{\Omega}}^{\frac{2n}{n-2}}<0$, then $\lambda_1(\Omega)<\lambda_*$ and there exist at least two solutions to \eqref{Eq1.1} for $\lambda\in (\lambda_1(\Omega),\lambda_*)$. Here $\phi^1_{\Omega}$ is the first eigenfunction of Laplacian in $\Omega $. \end{itemize} \item Suppose that assumptions (H1)-(H4) and (H6)-(H7) are satisfied. Then, there exists a continuum of solutions, $\mathcal{C}$, to \eqref{Eq1.2} in $\mathbb{R}^+\times C^1_0(\Omega)$ bifurcating from $\lambda_1(\Omega)$ and satisfying \begin{itemize} \item[(i)] $\Pi_{\mathbb{R}}\mathcal{C}=[0,\lambda_*]$ where $\lambda_1(\Omega)\leq\lambda_*<\lambda_1(\Omega^+)$. \item[(ii)] If $\phi(u)\sim C_2u^q$ when $u\to 0^+$ and if $\int_{\Omega}h{\phi^1_{\Omega}}^q<0$, then $\lambda_1(\Omega)<\lambda_*$ and there exist at least two solutions to \eqref{Eq1.2} for $\lambda\in (\lambda_1(\Omega),\lambda_*)$. \end{itemize} Furthermore, in both cases, there exists $(0, u_0) \in \mathcal{C} $ such that $u_0>0$. \end{enumerate} \end{theorem} We remark that for $n=2 $, we can even assert that the branch extends beyond 0, as done in \cite[Theorem 1.3]{AdGi}. If we consider star shaped domains as in \cite{ChLi2}, then we can relax the flatness condition as follows: \begin{itemize} \item[(H5c)] $c \mathop{\rm dist}(x, S_1)^{\beta-1}\leq |\nabla h(x)|\leq C\mathop{\rm dist}(x,S_1)^{\beta-1}$ for all $x\in\Omega^+$ with $c,C>0$ and $n-2 < \beta < n$ for $n > 4$. \end{itemize} \begin{theorem}\label{th2} Suppose that assumptions (H1)-(H5c) are satisfied and $\Omega$ is star shaped. Then the same conclusions as in Theorem \ref{th1}, (1), hold for $n > 4$. \end{theorem} Notice that we get at least one positive solution for both problems for $\lambda > 0 $, near zero and at least 2 solutions for some range of $\lambda$. We prove theorems \ref{th1} and \ref{th2} using the bifurcation theory of Rabinowitz (see \cite{Ra}). The crucial step here is to prove the existence of uniform a priori bound in $L^{\infty}(\Omega)$ of solutions to \eqref{Eq1.1} and \eqref{Eq1.2} independent of $\lambda$ in a compact set. Note that such a priori estimates are not obvious since the nonlinearity is critical and indefinite in sign in both equations \eqref{Eq1.1} and \eqref{Eq1.2}. This problem has been studied when $\Omega^0$ is empty and $h < 0$ near the boundary, in \cite{AdGi} for dimension 2 and in \cite{GiPrRa} for dimension 5 and above, using uniform bounds. Our aim here is to extend these results to the case of nonempty $\Omega^0$ and $h$ positive near the boundary and also to explore other possible flatness conditions. Ouyang \cite{TOu} has studied the bifurcation for equation \eqref{Eq1.1} without the uniform bound and flatness assumptions. Here we are able to conclude that $(0, u_0)$ lies in the branch for some $u_0 > 0$ because of the bound. Several results regarding a priori estimates are available concerning the case, $h$ positive, for the subcritical case (i.e. $u^p$ with $1 0$ near the boundary, the assumptions on the weight functions are suitably used to avoid the boundary blow up. Now, let us state the main results concerning a priori estimates for solutions to \eqref{Eq1.1} and \eqref{Eq1.2}: \begin{proposition}\label{est1} Let us assume $n > 4$, (H1)-(H5a) or (H5b) or for star shaped domains, (H5c). Let $\Lambda$ such that $0\leq\Lambda<\lambda_1(\Omega^+)$. Then for any solution $(\lambda,u)$ to \eqref{Eq1.1}, such that $0\leq \lambda\leq \Lambda$, we have \[ \|u\|_{L^{\infty}(\Omega)}\leq C(\Lambda ,n, h, \Omega). \] \end{proposition} Similarly for \eqref{Eq1.2}, we have the estimate: \begin{proposition}\label{est2} Let us assume $n=2$, (H1)-(H4) and (H6)-(H7). Let $\Lambda$ be such that $0\leq\Lambda<\lambda_1(\Omega^+)$. Then for any solution $(\lambda,u)$ to \eqref{Eq1.2}, such that $0\leq \lambda\leq \Lambda$, we have \[ \|u\|_{L^{\infty}(\Omega)}\leq C(\Lambda, h, \Omega). \] \end{proposition} Finally, let us mention that using the same ideas, we can also deal with \eqref{Eq1.1} and \eqref{Eq1.2} in the case of $\Omega=\mathbb{R}^n$. It has been done in \cite{AdGi}, \cite{GiPrRa} with $\Omega^0=\emptyset$. In these works, $h(x)\to -\infty$ when $|x|\to +\infty$. Then the problem becomes sublinear at infinity. So contrary to what we do in the present paper, no careful analysis near the boundary has to be done. But uniform decay at infinity has to be worked out thanks to the behaviour of $h$ at infinity. Now, to conclude this introduction, let us present the outline of this paper: In the second section, we give first a priori bound in $$ \Omega^-_{\delta}:=\{x\in\Omega^-: \mathop{\rm dist}(x,\Gamma\cup\Omega^0) \geq \delta\}. $$ The estimate in this region is performed by a $L^q$-estimate and hypoellipticity arguments. The third section, deals with a neighborhood of $\Gamma$ and $\Omega^0$. We use the moving plane method which allows us to give a priori bound in a neighborhood of $\Gamma$ for both problems as in \cite{GiPrRa} and \cite{AdGi} (see also \cite{BiGi}). After that, by exhibiting a supersolution, we get an a priori estimate in a neighborhood of $\Omega^0$. Here the fact that $\lambda$ remains uniformly below to $\lambda_1(\Omega^0)$ is crucial for getting the existence of the supersolution. Section 4 is on a priori bound in $$ \Omega^+_{\delta,\eta}:=\{x\in\Omega^+_{\delta}: \mathop{\rm dist}(x,\partial\Omega)\geq \eta\} $$ for some ($\delta, \eta >0$). As in \cite{GiPrRa}, we extend the blow up analysis of \cite{Yl} for $\lambda \geq 0$, indicating the proofs with either of the flatness assumptions (H5a), (H5b) or (H5c). Section 5 is concerned with $\partial \Omega$. Concerning \eqref{Eq1.1}, the most difficult case is when $h$ is positive near the boundary. In this case, we use a blow up method as in \cite{ChLi2} to estimate solutions close to the boundary. If $h$ is non positive, we use the maximum principle. This finalizes the proof of Propositions \ref{est1} and \ref{est2}. In the final section, we prove the main results. We would like to point out that this result holds also in the pure superlinear case (ie. $h>0$ in $\overline{\Omega}$). \section{A priori bound in $\Omega^-_{\delta}$} Here we obtain a priori bounds for the solution $u$ of \eqref{Eq1.1} and \eqref{Eq1.2} in the region $\Omega^{-}_{\delta}$. Note that $\Omega^{-}_{\delta}$ contains a neighborhood of $\partial\Omega$ if $h<0$ near $\partial\Omega$. The following $L^p$ estimate is crucial, which in fact is true in both sets $\Omega^{-}$ and $\Omega^{+}$. \begin{proposition} \label{prop:Lp-Estimate} Let $x_0 \in \Omega^{\pm}$, $\epsilon > 0$, $\Lambda\in\mathbb{R}^+$, and $B_{\epsilon} (x_0) \subset \subset \Omega^{\pm}$. Assume that $\lambda\leq\Lambda$. \begin{itemize} \item[(i)] For $n = 2$, if $u$ is a solution of \eqref{Eq1.2} then \begin{equation} \label{e2'} \int_{B_{\frac{\epsilon}{2}}(x_0)}e^u\,\leq\,C( \epsilon, h,\Lambda). \end{equation} \item[(ii)] For $n \geq 3$, if $u$ is a solution of \eqref{Eq1.1}, then there exists $C = C(\epsilon,\Lambda)$ such that \begin{equation} \label{e4} \int_{B_{\frac{\epsilon}{2}} (x_0) } u^{\frac{n+2}{n-2}} dx \leq \Big( \frac{C}{ \inf_{B_{\epsilon}} |h| }\Big)^{\frac{n+2}{4}}. \end{equation} \end{itemize} \end{proposition} This estimate follows by multiplying by the first eigenfunction of Laplacian in a ball and using integration by parts. The details are in \cite{AdGi} and \cite{GiPrRa} respectively. We present the proof for \ref{Eq1.1} here. The proof for the other equation is similar. For a bound for $u$ in $\Omega_{\delta}^-$, we need to consider 2 cases: \begin{itemize} \item[(i)] Either $h<0$ near $\partial\Omega$, or \item[(ii)] $h>0$ near $\partial\Omega$. \end{itemize} In case (i), let us define a $\delta$ neighborhood of $\partial\Omega$ $$ G:=\{x\in\Omega_{\delta}^-:\mathop{\rm dist}(x,\partial\Omega)\leq\delta\}. $$ For $\delta$ small enough, by (H3) and (H4), $G\subset\Omega^-_{\delta}$. Let $A:= \{ x \in \Omega_{\delta}^-: -\Delta u(x) < 0\}$. We split now the domain into three sets $$ \Omega_{\delta}^- = (\Omega_{\delta}^- \setminus G) \cup (G \setminus A ) \cup (G \cap A ). $$ We will get the apriori estimate in the first set using the earlier integral estimate and in the second one, a pointwise estimate and then in the third set via maximum principle and the previous estimates. For $x \in \Omega_{\delta}^-\setminus G$, there exists a ball $B_{\delta/2}(x)\subset\Omega_{\frac{\delta}{2}}^-$ and the integral estimate (\ref{e4}) hold for $u$ in $B_{\delta/4}(x)$. Then we use the following Lemma \cite[Lemma 9.20]{GT}. \begin{lemma}\label{gitr} Let $u\in W^{2,n}(\Omega)$ with $Lu\geq f$ where $L$ is a strictly elliptic second order operator and $f\in L^{n}(\Omega)$. For all $B=B_{\epsilon}(y)\subset\Omega$ and $q>0$, we have \begin{equation}\label{e6} \sup_{B_{\frac{\epsilon}{2}}(y)}u\leq C(n,q,\epsilon) \Big(\Big(\int_B(u^+)^q\Big)^{\frac{1}{q}} +\|f\|_{L^n(B)}\Big). \end{equation} \end{lemma} We combine this estimate for $f=0$, $q = \frac{n+2}{n-2}$ and $L=\Delta+\lambda$ in the ball $B_{\frac{\delta}{2} }(x) \subset \Omega_{\delta/2}^-$ (since $x\in\Omega^-_{\delta}$), together with the estimate (\ref{e4}) to conclude that \begin{equation} \label{eq:Celta} \sup_{B_{\frac{\delta}{8}}(x)} u \leq C(n, \lambda,\delta) \Big\{ \frac{1}{\inf_{B_{\delta/2} (x)} |h| } \Big\}^{(n-2)/4}. \end{equation} Note that here we can take any $\lambda < \Lambda $. Thus we have if $x \in \Omega_{\delta}^- \setminus G $, \begin{equation} \label{-1} u(x) \leq C(n, \Lambda,\delta) \Big \{ \frac{1}{\inf_{\Omega_{\delta/2}^-} |h| } \Big \}^{(n-2)/4} . \end{equation} In the case $ x \in G \setminus A$, $$ 0 \leq -\Delta u(x) = \lambda u(x)+h(x)u^{\frac{n+2}{n-2}}(x), $$ and hence $$ - h(x)u^{\frac{n+2}{n-2}}(x) \leq \lambda u(x). $$ Since $u(x) > 0$, we have the {\em pointwise estimate} \begin{equation}\label{e5} u(x) \leq \Big (\frac{\lambda}{\inf_{\Omega_{\delta}^-} |h|} \Big )^{(n-2)/4} \quad\text{for all } x \in G \setminus A. \end{equation} Using the above estimates and recalling that $u=0$ on $\partial \Omega $, we have for points on $\partial (G \cap A)$, \begin{equation} \label{-2} u(x) \leq M , \quad M = \max \big\{C(n, \lambda,\delta) \{ \frac{1}{\inf_{\Omega_{\delta/2}^-} |h| } \}^{(n-2)/4}, (\frac{\lambda}{\inf_{\Omega_{\delta}^-} |h|} )^{(n-2)/4} \big\}. \end{equation} Define $c(x):= \lambda + h(x) u(x)^{4/n-2}$ and consider the equation \begin{equation}\label{00} \Delta v + c(x) v \geq 0 \quad \text{in } ( G \cap A). \end{equation} Note that for $x \in A$, $c(x) < 0$ and that $u - M $ is a solution of (\ref{00}). Hence, by the weak maximum principle (\cite[Theorem 9.1]{GT} with $ f\equiv 0$), we have $$ u(x)- M \leq 0 \quad\text{in } (G \cap A). $$ Combining all the cases, we have the local estimate in this case. Now let us consider the case (ii), $h>0$ near $\partial \Omega$. Then, by (H3) since $\Gamma \subset \Omega$, for $\delta$ small, $\mathop{\rm dist}(\Omega_{\delta/2}^-,\partial\Omega)>\delta$. Therefore a straightforward application of Lemma \ref{gitr} yields (\ref{eq:Celta}). Thus we have the following local estimate in $\Omega_{\delta}^-$. \begin{proposition}\label{o-} Let $ 0 \leq \lambda \leq \lambda_1(\Omega^+)$. Assume (H1)--(H3). \begin{itemize} \item[(i)] For $u$, a solution of \eqref{Eq1.2} and for $1 < p' < \infty$, \begin{equation}\label{eo-2} \sup_{\Omega_{\delta}^-} u(x) \leq C( \lambda_1(\Omega^+),\delta) \Big( \frac{1}{\inf_{\Omega_{\delta/2}^- } |h|}\Big)^{1/(p'-1)}. \end{equation} \item[(ii)] For $u$, a solution of \eqref{Eq1.1} \begin{equation}\label{eo-} \sup_{\Omega_{\delta}^-} u(x) \leq C(n,\lambda_1(\Omega^+), \delta) \Big( \frac{1}{\inf_{\Omega_{\delta/2}^-} |h|}\Big)^{(n-2)/4}. \end{equation} \end{itemize} \end{proposition} Note that since $\inf_{\Omega^-_{\delta/2}}|h|>0$, from Proposition \ref{o-}, we get uniform a priori bounds in $\Omega^-_{\delta}$. \section{A priori bound in a neighborhood of $\Gamma$ and $\Omega^0$} Let us start by giving a priori bound in a neighborhood of $\Omega^0$. Let $\Omega_{\delta}^0$ be a smooth $\delta$-neighborhood of $\Omega^0$. By (H4)-1 and (H4)-2 and the results in the second section (bounds in $\Omega^-_{\delta}$), solutions to $\eqref{Eq1.1}$ and \eqref{Eq1.2} are uniformly bounded in $\partial\Omega_{\delta}^0$ by a constant $M$. Now, Let $\psi$ the solution to \begin{equation} \begin{gathered} - \Delta \psi =\Lambda \psi \quad \text{in } \Omega_{\delta}^0 \\ \psi > 0 \quad \text{in } \Omega_{\delta}^0; \quad \psi= M\quad \text{on } \partial\Omega_{\delta}^0 \end{gathered} \end{equation} which exists if $\Lambda<\lambda^1(\Omega^0)$ and $\delta$ small. By maximum principle and for $\lambda\leq \Lambda$, any solution $(\lambda,u)$ to $\eqref{Eq1.1}$ and \eqref{Eq1.2} satisfies $u\leq \psi$ in $\Omega_{\delta}^0$. Now, the estimates in a $\delta$-neighborhood of $\Gamma$, $\Gamma_{\delta}$, are similar to step 2 in \cite{BiGi} (for solutions to \eqref{Eq1.1} just replace $p$ by $\frac{n+2}{n-2}$). Note that for solutions to \eqref{Eq1.2}, we use the assumption (H7) to carry out the moving plane method. Precisely, to show that (taking the same notations as in step 2 in \cite{BiGi}) $\frac{\partial f}{\partial x_1}\leq 0$ we use that $\phi(u)+\phi'(u)\geq 0$ and bounded. \section{ A Priori Bound in $\Omega^+_{\delta,\eta}$} We start with \eqref{Eq1.1}. This part is done in \cite{GiPrRa} and used the blow up analysis in \cite{Yl}. Precisely, let a sequence $(\lambda_i,u_i)$ be solutions of \eqref{Eq1.1}, such that $0\leq\lambda_i\leq\lambda_1(\Omega^+)$ and a sequence of local maxima ${x_i}\in \Omega_{\delta}^+$ of $u_i$, such that $u_i (x_i) \to \infty$ as $i \to \infty$. By the a priori bounds on $\Omega_{\delta}^-$, $\Gamma$ and $\Omega^0)$ obtained in the earlier 2 sections, and also in view of the bound to be obtained in the next section in a neighbourhood of $ \partial \Omega $, $\{u_i\}$ remains uniformly bounded on the boundary of $\Omega^{+}_{\delta, \eta}$. Thus ${x_i}$ has to converge to some point in the interior of $\Omega^+_{\delta,\eta}$. Note that the a priori bound near the boundary is independent of the results of this section. We postpone that because we need to use some results from this section, in particular, Proposition \ref{P1}. In the first subsection we recall the standard blow up argument to analyse $u_i$, in a small neighbourhood of $x_i$ and various local estimates required later on. In the second subsection, we use these estimates to prove that a blow up point of ${u_i}$ is necessarily a critical point of $h$. This motivates the flatness assumptions at the critical points of $h$. Using this assumption, we analyse the nature of the blow up points and show that in fact ${u_i}$ does not blow up, i.e the sequence $\{u_i\}$ is uniformly bounded. \subsection{Blow up points of $\{u_i\}$} We start by recalling several definitions and propositions which are extensions of some results in \cite{Yl} to the case of $ \lambda > 0$. As the details are in \cite{GiPrRa}, we give here only an outline, giving details only when new ideas are involved. The following is a standard result in the blow up analysis (see for example \cite{SC}). \begin{proposition} \label{P1} Suppose that $h \in C^1 ( \Omega^{+}_{\delta})$ and there exist $A_1$ and $A_2$ such that in $ \Omega^{+}_{\delta}$, $$ h (x) \geq \frac{1}{A_1}, \quad \|\nabla h (x)\| \leq A_2 . $$ Then for every $0 < \varepsilon <1, \; R>1$, there exist positive constants $C_0$ and $C_1$ depending on $A_1, \ A_2, \varepsilon, R, \lambda $ and $n$ such that if $v$ is a positive solution of \begin{equation} \label{Eq4.2} - \Delta v (x) = \lambda v (x) + h (x) v^{\frac{n+2}{n-2}}, \quad v>0 \end{equation} with $\max_{B} v > C_0 $, then there exists a finite number $k= k (v)$ and a set $ S (v ,C_0) = \{x_1,\ldots, x_k\} \subset \Omega^{+}_{\delta}$ such that \begin{itemize} \item[(i)] $x_j$ are the local maxima of $v$ and for $\mu_j = v (x_j)^{-{\frac{2}{n-2}}}, \left\{B_{R \mu_j} (x_j)\right\}_{1 \leq j \leq k}$ are disjoint balls and $$\|v (x_j)^{-1} v (x_j +\mu_j x) - \delta_j (x) \|_{C^2 (B_{2R} (0))} < \varepsilon, $$ where $$ \delta_j (x) = (1+h_j |x|^2)^{\frac{2-n}{2}}\quad\text{with } h_j = (n(n-2))^{-1} h (x_j) $$ is the unique solution of \begin{gather*} \Delta \delta_j +h_j \delta_j^{\frac{n+2}{n-2}} = 0 \quad \text{in } \mathbb{R}^n,\\ \delta_j > 0 \quad \text{in } \mathbb{R}^n, \quad \delta_j (0) = 1, \end{gather*} \item[(ii)] $v (x) \leq C_1 (\mathop{\rm dist}(x, S))^{-(\frac{n-2}{2})}, x \in \Omega^{+}_{\delta} $. \end{itemize} \end{proposition} The above Proposition, in particular (ii), motivates the definition of an isolated blow up point. \begin{definition} \label{def1} \rm A point $x_0 \in \Omega'$ is called an isolated blow up point of $\{u_i\}$, solutions of \eqref{Eq1.1}, if there exists $0 <\bar{r} < dist (x_0,\partial \Omega')$ and $C>0$ and a sequence $\{x_i\}$ tending to $x_0$, such that $x_i$ is a local maximum of $\{u_i\}, u_i (x_i) \to \infty$ and $$ u_i (x) \leq C |x - x_i|^{-(\frac{n-2}{2})} \quad \forall x \in B_{\bar{r}} (x_0)\backslash\{x_i\}. $$ \end{definition} Since we will be interested in the blow up points staying away from each other, we also need to introduce the definition of a simple isolated blow up point. \begin{definition} \label{def2} \rm A point $ x_0 $ is an isolated simple blow up point of $\{u_i\}$, solutions of \eqref{Eq1.1}, if it is an isolated blow up point such that for some $\rho >0$ (independent of $i$), $\tilde{v}_i$ has precisely one critical point in $(0, \rho)\; \forall$ large $i$, where $$ {\tilde{v}}_i (r)= r^{\frac{n-2}{2}} \overline{v}_i (r), \quad \overline{v}_i (r) = \frac{1}{|\partial B_r|} \int_{\partial B_r (x_i)} u_i, \quad r >0. $$ \end{definition} The following is a corollary of Proposition \ref{P1}. \begin{corollary}\label{C1} Let $x_0$ be an isolated blow up point of $ \{u_i\}$. Then one can choose $R_i \to \infty$ first and then $\varepsilon_i \to 0^+$ depending on $R_i$ and a subsequence $\{u_i\}$ so that \begin{itemize} \item[(i)] $r_i = \frac{R_i}{(u_i (x_i))^{\frac{2}{n-2}}} \to 0$ and $x_i$ is the only critical point of $u_i (x)$ in $|x -x_i| < r_i$. \item[(ii)] ${\tilde{v}}_i (r)$ has a unique critical point in $0 < r < r_i$. \end{itemize} \end{corollary} Another important result, we will use in the following, is the Harnack inequality \begin{lemma}[A Harnack inequality] \label{harnack} Let $h$ satisfy \begin{equation} \frac{1}{A_1} \leq h (x) \leq A_1 \quad \forall x \in \Omega_\delta^+ \label{Eq4.5} \end{equation} and $\{u_i\}$ satisfy \eqref{Eq1.1} having $0$ as an isolated blow up point. Then for any $0 < r <\frac{\bar{r}}{3}$, with $\bar{r} $ as in Definition \ref{def1}, we have the Harnack inequality \begin{equation} \max_{B_{2r} \setminus B_{r/2}} \; u_i (y) \leq C \min_{B_{2r}\setminus B_{r/2}} \; u_i (y) \label{Eq4.6} \end{equation} with a uniform $C = C (n, \lambda, \|h\|_{L^\infty (\Omega^+_\delta)})$. \end{lemma} The proof of this lemma follows on the same lines as in \cite{Yl}, \cite{CeGr}. Now, we look for lower and upper bounds for $u_i$, in a fixed neighbourhood of the blow up point. The arguments for the lower bound are as in \cite{Yl} (section 2 there). For the upper bound, we need to exploit specifically the extra linear term in our case, as in \cite{GiPrRa}. \begin{proposition}\label{P2} Assume $B_2(0)\subset\Omega^+_\delta$ and \begin{equation}\label{P2a} A_1 \geq h(x) \geq \frac{1}{A_1},\quad \|\nabla h(x) \| \leq A_2 \quad \forall x \in B_2 \end{equation} for some positive constants $A_1$, $A_2$. Let $u_i$ be solutions of \eqref{Eq1.1} and $x_i \to 0$ be an isolated blow up point with \begin{equation}\label{P2b} u_i(x) \leq \frac{A_3}{|x-x_i|^{\frac{n-2}{2}}} \quad \text{for all } x \in B_2\backslash\{x_i\}. \end{equation} Then there exists a positive constant $C= C(n,\lambda_0,A_1,A_2,A_3)$, such that up to a subsequence, \begin{equation}\label{P2c} u_i(x) \geq C u_i(x_i) ( 1 + h_i u_i(x_i)^{\frac{4}{n-2}}|x-x_i|^2 )^{(2-n)/2} \quad\text{ for all } |x- x_i| \leq 1, \end{equation} where $h_i$ is as defined in Proposition \ref{P1}. In particular, for any $e \in \mathbb{R}^n$, $|e| =1$, we have \begin{equation} \label{P2d} u_i(x_i +e )\, \geq \, C^{-1} u_i(x_i)^{-1}. \end{equation} \end{proposition} \begin{proposition}[Upper bound] \label{P3} Let $h$ and $\{u_i\}$ satisfy the conditions in Proposition \ref{P2}. Also, assume that $x_i \to 0$ is an isolated simple blow up point as defined in Definition \ref{def2}. Then there exists a positive constant $C = C(n, \lambda_0, A_1,A_2,A_3, \rho)$ such that \begin{equation}\label{P3a} u_i(x) \, \leq \,C u_i(x_i)^{-1} |x-x_i|^{2-n} \quad \text{for all } \; 0< |x-x_i| \leq 1. \end{equation} In particular, $u_i(x_i+e)u_i(x_i)=O(1)$, where $e$ is a unit vector in $\mathbb{R}^n$. \end{proposition} The following lemma is crucial in our analysis. \begin{lemma}\label{L3} Under the assumptions of Proposition \ref{P3}, let $\{ (\lambda_i, u_i) \} $ be a sequence of solutions of \eqref{Eq1.1} and $x_i \to 0$ be an isolated simple blow up point. Then upto a subsequence, we have \begin{itemize} \item[(i)] if $\lambda_i $ goes to zero, then $$ u_i (x) u_i (x_i) \to w(x) = \frac{a}{|x|^{n-2}} + g(x) $$ where $a = h_i^{-(n-2)/2}$ and $ g$ is some harmonic function. \item[(ii)] if $\lambda_i $ goes to $ \tilde{\lambda} > 0 $, then $$ u_i (x) u_i (x_i) \to w(x) = \frac{\alpha C_n}{|x|^{n-2}} + E(x) + \varphi(x) $$ where $G(x) = \alpha C_n |x|^{2-n} + E(x) $ is the unique solution in the sense of distribution for the equation \begin{gather*} -\Delta G = \lambda G + \alpha \delta_0 \quad\text{in }B_{\sigma_1}(0)\\ G = 0 \quad\text{on } \partial B_{\sigma_1}(0) \end{gather*} and $\varphi$ is the unique $C^2$ solution of the boundary value problem \begin{gather*} -\Delta \varphi = \lambda \varphi \quad\text{in }B_{\sigma_1}(0)\\ \varphi = w \quad\text{on } \partial B_{\sigma_1}(0). \end{gather*} Here $\sigma_1$ is sufficiently small such that $\lambda < \lambda_1(B_{\sigma_1}(0))$. \end{itemize} \end{lemma} The proofs are in \cite{GiPrRa}. There the following lemma is used to handle the linear term. This imposes a restriction $ n > 4$. Now, we fix $e\in\mathbb{R}^n$ such that $|e|=1$. As in \cite{CeGr} (see Proposition 3.5), we have the following result. \begin{lemma}\label{estcegr} Let us suppose $\{(\lambda_i, u_i)\}$ is a sequence of solutions of \eqref{Eq1.1} and $\bar{x}=0$ is an isolated and simple blow up point. Suppose also that $n > 4$. Then, there exists a positive constant $C=C(n,h, \rho)$ such that \begin{equation}\label{cest} \lambda_iu_i(x_i)^{\frac{2(n-4)}{n-2}}\leq\, Cu_i(x_i)^2u_i(x_i+e)^2+o(1) \, . \end{equation} \end{lemma} \subsubsection{Nature of Blow up points for solutions to \eqref{Eq1.1}} We need the following Identity. \begin{lemma}[Pohozaev Identity] \label{poh} Let $v$ be a $C^2$ solution of \eqref{Eq1.1}. Then for any $B_\sigma \subset \Omega^+_{\delta}$, \begin{align*} \int_{\partial B_\sigma} B (\sigma, x, v,\nabla v) &= \Big\{ \lambda \int_{B_\sigma} v^2 - \frac{\lambda}{2} \int_{\partial B_\sigma} \sigma v^2 \Big\} + \frac{n-2}{2n} \int_{B_\sigma} (x \cdot \nabla h) v^{\frac{2n}{n-2}}\\ &\quad - \frac{\sigma(n-2)}{2n} \int_{\partial B_\sigma} h (x) v^{\frac{2n}{n-2}}, \end{align*} where $$ B (\sigma, x, v, \nabla v):= - \frac{(n-2)}{2} v \frac{\partial v}{\partial \nu} - \frac{ \sigma}{2} |\nabla v|^2 + \sigma \big|\frac{\partial v}{\partial \nu} \big|^2 $$ and $\nu$ denotes the unit outer normal vector field on $\partial B_{\sigma}$. \end{lemma} For a proof see for example \cite{Yl}, \cite{GiPrRa}. The earlier estimates and Pohozaev identity can be used to derive various conclusions about the possible blow up points of $\{ u_i\}$. In particular, the following result is often used. \begin{lemma}\label{C2} For $u(x) = \frac{a}{|x|^{n-2}} + b(x)$ where $a > 0$ and $b(x)$ is a nonnegative differentiable function, with $b(0) > 0 $, we have $$ B (\sigma, x, u, \nabla u) \, < \, 0, $$ on $\partial B_{\sigma}$ for all $\sigma$ small. \end{lemma} The proof follows from direct computations. After suitable rescaling of the solutions, using the above identities, one can prove that in fact that the blow up points are simple, isolated and they have to be the critical points of $h$, as in \cite{Yl,GiPrRa}. \begin{proposition}\label{nbp} Suppose that (H1)-(H5) are satisfied. Then any isolated blow up point is simple and is a critical point of $h$. \end{proposition} \begin{proposition}\label{P5'} Suppose that (H1)-(H5) are satisfied.The blow up points of $\{ u_i \}_i $ are isolated: More precisely, for $\varepsilon >0$ and $R >1$, there exists some positive constant $r^\ast = r^\ast (n, \varepsilon, R,A_1, c_1, c_2,d$, modulus of continuity of $\nabla h)$ such that for any solution $u_i$ with $\max_{\Omega'} u_i > C^\ast$, we have $$ |q_{l} - q_{j}|\geq r^\ast \quad \forall 1 \leq l \neq j \leq k, $$ where $q_{l} = q_{l} (u_i), \quad k = k (u_i)$ are as in Proposition \ref{P1}. \end{proposition} \subsubsection{A priori estimates for solutions of \eqref{Eq1.1}} \begin{proposition} \label{P5} Assume (H1)-(H4). Further either (H5a) or (H5b) holds or (H5c) holds for $\Omega$ star shaped. Let $\{(\lambda_i, u_i)\}_i $ be a sequence of solutions of \eqref{Eq1.1} with $0\leq\lambda_i \to \tilde{\lambda}$. Then, $\{ u_i \}_i $ is uniformly bounded in $L^{\infty}(\Omega^+_{\delta,\eta})$. \end{proposition} \proof Let us first consider the case when (H5a) holds. \\ (i) If $\tilde{\lambda} > 0$, the analysis follows from Proposition 5.5 (iii) of \cite{GiPrRa} . Using Pohozaev identity in a ball around a blow up point, a contradiction is derived there. So $\tilde{\lambda}$ cannot be positive. \noindent(ii) If $\tilde{\lambda} =0$ and $n \geq 5$, we get a contradiction by using Pohozaev identity as in the Appendix in \cite{GiPrRa}. Combining both, we conclude that the solutions are uniformly bounded, if (H5a) holds. Below we recall the proof for $\tilde{\lambda}=0$ and $n > 4$ when (H5a) holds because some estimates will be needed for the other cases. We will follow the arguments in \cite[Theorem 4.4]{Yl}, adapted here for the bounded domain $\Omega^+_{\delta,\eta}$. In the proof, we need the following Pohozaev identity: If $u$ is a $C^2$-solution to $-\Delta u=\lambda u+h(x)u^{\frac{n+2}{n-2}}$ in $B_{\sigma}$ for some $\sigma >0$, then \begin{equation}\label{numb} \int_{B_{\sigma}}(\nabla h)u^{\frac{2n}{n-2}} =\frac{2n}{n-2}\int_{\partial B_{\sigma}}(\frac{\partial u}{\partial \nu}\nabla u-\frac{1}{2}|\nabla u|^2\nu) +\int_{\partial B_{\sigma}}\frac{\lambda}{2}u^2\nu+\int_{\partial B_{\sigma}}hu^{\frac{2n}{n-2}}\nu \end{equation} where $\nu$ is the unit outward normal. Let $\{y_i\}\in \Omega^+_{\delta,\eta}$. Without loss of generality, we can assume that $y_i\to 0$. Then, for a small ball $B_{\sigma}$ around $y_i$, by the above Pohozaev identity, we have \[ \int_{B_{\sigma}}\nabla h(x)(u_i(x))^{\frac{2n}{n-2}}=I_1+I_2+I_3 \] where \begin{gather*} I_1=\frac{2n}{n-2}\int_{\partial B_{\sigma}}(\frac{\partial u_i}{\partial \nu}\nabla u_i-\frac{1}{2}|\nabla u_i|^2\nu),\\ I_2=\frac{\lambda_i}{2}\int_{\partial B_{\sigma}}u_i^2\nu,\quad I_3=\int_{\partial B_{\sigma}}h(x)(u_i(x))^{\frac{2n}{n-2}}\nu . \end{gather*} We will estimate $I_k$'s as follows: First, we observe that \[ |I_1|\leq C\int_{\partial B_{\sigma}}|\nabla u_i|^2. \] Then, we need to evaluate $\int_{\partial B_{\sigma}}|\nabla u_i|^2$ for a suitable value of $\sigma$. For this, let $A: =B_{\frac{1}{2}}\backslash B_{\sigma_1}$ such that $\sigma_1 <\frac{1}{2}$. Let $\eta_i$ a cut off function such that \[ \eta_i(x)=\begin{cases} 1 & \text{if } x\in A_i=\{ x,\sigma_1+\epsilon_i\leq |x|\leq \frac{1}{2}-\epsilon_i\}\\ 1-\frac{|x|}{\epsilon_i}& \text{if } x\in A\backslash A_i \end{cases} \] where $\epsilon_i:=u_i(y_i)^{-1}$. Now, multiplying the equation \eqref{Eq1.1} by $\eta_i u_i$, we get \[ \int_{A}(\nabla u_i\cdot\nabla(\eta_i u_i))=\lambda_i\int_{A}u_i^2\eta_i+\int_A hu_i^{\frac{2n}{n-2}}\eta_i. \] It follows that \begin{gather} \begin{aligned} \int_{A_i}|\nabla u_i|^2 &\leq \int_{A}|\nabla u_i|^2\eta_i \\ &\leq \lambda_i\int_{A}u_i^2+\int_A hu_i^{\frac{2n}{n-2}} +\frac{1}{2}|\int_{A\backslash A_i}\nabla(u_i^2)\cdot\nabla\eta_i|, \end{aligned}\label{cccest} \\ \label{Su2} \int_A hu_i^{\frac{2n}{n-2}}\leq\frac{C}{(u_i(y_i))^{\frac{2n}{n-2}}}, \\ \begin{aligned} |\int_{A\backslash A_i}\nabla(u_i^2)\cdot\nabla\eta_i| &\leq C\Big(\int_{A\backslash A_i}|\nabla(u_i^2)|^2\Big)^{1/2} \times\Big(\int_{A\backslash A_i}|\nabla\eta_i|^2\Big)^{1/2} \\ &\leq \frac{C}{(u_i(y_i))^2}\times \frac{1}{(u_i(y_i))^{n-1}} =\frac{C}{(u_i(y_i))^{n+1}}. \end{aligned}\label{ccest} \end{gather} Hence, from (\ref{cccest}), (\ref{Su2}) and (\ref{ccest}), we get \[ \int_{A_i}|\nabla u_i|^2\leq \begin{cases} \frac{C}{(u_i(y_i))^{\frac{2n}{n-2}}}&\text{for }n\geq 6,\\ \frac{C}{(u_i(y_i))^{\frac{8}{3}}} &\text{for }n=5. \end{cases} \] Now, taking $\sigma_i\in [\sigma_1+\epsilon_1,\frac{1}{2}-\epsilon_1]$ such that \[ \int_{\partial B_{\sigma_i}}|\nabla u_i|^2 =\min_{\sigma\in[\sigma_1+\epsilon_1,\frac{1}{2}-\epsilon_1]} \int_{\partial B_{\sigma}}|\nabla u_i|^2, \] we have \[ \int_{\partial B_{\sigma_i}}|\nabla u_i|^2\leq \begin{cases} \frac{C}{(u_i(y_i))^{\frac{2n}{n-2}}}\times\frac{1}{(\frac{1}{2} -\sigma_1-2\epsilon_i)}&\text{for }n\geq 6,\\ \frac{C}{(u_i(y_i))^{\frac{8}{3}}}\times\frac{1}{(\frac{1}{2} -\sigma_1-2\epsilon_i)}&\text{for }n=5 \end{cases} \] Therefore, \begin{equation} \label{new1} |I_1|, |I_3| \leq \begin{cases} \frac{C}{(u_i(y_i))^{\frac{2n}{n-2}}} &\text{for }n\geq 6,\\ \frac{C}{(u_i(y_i))^{\frac{8}{3}}} &\text{for }n=5 \end{cases} \end{equation} Also we have for all $n > 4 $, $$ |I_3| \leq \frac{C}{(u_i(y_i))^{2n/(n-2)}}. $$ Then \[ \int_{B_{\sigma}}\nabla h(x)(u_i(x))^{\frac{2n}{n-2}} \leq \begin{cases} \frac{C}{u_i(y_i)^{\frac{2n}{n-2}}}&\text{for }n\geq 6,\\ \frac{C}{(u_i(y_i))^{\frac{8}{3}}}&\text{for }n=5 \end{cases} \] Now, we follow the arguments in \cite[Theorem 4.4 and Corollary 4.1]{Yl} to prove \begin{itemize} \item[Step 1:] $|y_i|=O(\frac{1}{u_i(y_i)^{\frac{2}{n-2}}})$ so that $y_i u_i(y_i)^{\frac{2}{n-2}}=\xi_i\to\xi$, \item[Step 2:] Multiplying (\ref{numb}) by $u_i(y_i)^{\frac{2(\beta-1)}{n-2}})$, and using estimates on $I_i$ and rescaling arguments, we get $\int_{\mathbb{R}^n}\nabla h(z+\xi)\frac{dz}{(1+k^2|z|^2)^n}=0$ which contradicts (H5a). \end{itemize} Let us consider the case when (H5b) holds. In this case, we use Lemma \ref{poh}, \begin{align*} \int_{\partial B_\sigma} B (\sigma, x, v,\nabla u_i) &= \Big\{ \lambda_i \int_{B_\sigma} u_i^2 - \frac{\lambda_i}{2} \int_{\partial B_\sigma} \sigma u_i^2 \Big\} + \frac{n-2}{2n} \int_{B_\sigma} (x \cdot \nabla h) u_i^{\frac{2n}{n-2}}\\ &\quad - \frac{\sigma(n-2)}{2n} \int_{\partial B_\sigma} h (x) u_i^{\frac{2n}{n-2}}, \end{align*} from which together with (H5b), $\sigma<\sigma_0$ and $\lambda_i\geq 0$ it follows that \begin{equation} \label{h5b} \begin{aligned} \int_{B_\sigma} (x \cdot \nabla h) u_i^{\frac{2n}{n-2}} &\leq -\frac{2n}{n-2}\int_{\partial B_\sigma}B (\sigma, x, v,\nabla u_i) +\sigma\int_{\partial B_\sigma} h (x) u_i^{\frac{2n}{n-2}}\\ &\quad +\frac{\lambda_i}{2} \int_{\partial B_\sigma} \sigma u_i^2. \end{aligned} \end{equation} Then, from (\ref{Su2}) and (\ref{h5b}) we get \begin{equation} \label{h5b2} \begin{aligned} \int_{B_\sigma} |x|^{\beta} u_i^{\frac{2n}{n-2}} &\leq \int_{B_\sigma} (x \cdot \nabla h) u_i^{\frac{2n}{n-2}}\\ &\leq K_1\int_{\partial B_\sigma}u_i|\nabla u_i|+K_2\int_{\partial B_\sigma}|\nabla u_i|^2+O(u_i(y_i)^{-\frac{2n}{n-2}}). \end{aligned} \end{equation} The two boundary integrals can be estimated like $I_1$ in (\ref{new1}). Hence, we get for $ n> 5$, \begin{equation} \label{h5b4} \int_{B_{r_i}} |x|^{\beta} u_i^{\frac{2n}{n-2}}\leq O(u_i(y_i)^{-\frac{2n-2}{n-2}}). \end{equation} Now using a change of variable and using Proposition (\ref{P1}), (i), we see that the left hand side integral is positive but the right hand side goes to 0, since $\beta 0$ on $\partial \Omega $. Furthermore $ |\nabla w|^2 > 0$ on $\partial \Omega $ . Hence at least for some $i$, $$ I = \int_{\partial B_{\sigma} (p_i) } \Big(\frac{1}{2}|\nabla w|^2 (x \cdot \nu) -\frac{n-2}{2} \frac{\partial w}{\partial \nu} w - \frac{\partial w}{\partial \nu} (x \cdot \nabla w)\Big) > 0. $$ Now we apply Pohozaev identity on $B_{\sigma} (p_i) $ and multiply by $u_i(y_i)^2$ and pass to the limit to get $$ I = \lim (u_i^2 (y_i) \int_{B_{\sigma}} (x \cdot \nabla h) u_i^{\frac{2n}{n-2}} \, ) $$ The same calculations as in the proof of \cite[Proposition 5.5 (iii)]{GiPrRa} show that \begin{align*} u_i^2 \int_{B_{\sigma}} (x \cdot \nabla h) u_i^{\frac{2n}{n-2}} & \leq u_i^2 \int_{B_{\sigma}} |x|^{\beta} u_i^{\frac{2n}{n-2}} \\ & = \frac{1}{u_i^{\frac{2\beta}{n-2}-2}} \Big(\int_0^{R_i} \frac{r^{\beta + n - 1}}{(1+(h_i)^2 r^2)^{2n}} dr + o(1) \Big) \to 0 \end{align*} if $n-2 < \beta$. Thus for $\lambda_i$ approaching to 0, the solutions cannot blow up and hence are uniformly bounded. These prove the proposition \ref{est1}. \subsection{A priori bound for solutions to \eqref{Eq1.2} in $\Omega^+_{\delta,\eta}$} In the following, we assume (H1)-(H4), (H6), $n=2$, $0\leq\lambda\leq\Lambda$. We suppose also that {\it the solutions to \eqref{Eq1.2} are uniformly bounded on $\partial\Omega^+_{\delta,\eta}$}. These bounds are to be proved in the next section independent of the bounds obtained in this section. We recall the following result from \cite{BrMe}. \begin{theorem}[Blow-up analysis (Brezis-Merle)]\label{T1'} Assume $u_n$ a sequence of solutions to \begin{equation}\label{e0} -\Delta u_n= V_n(x) e^{u_n}\quad \text{in }\Omega \end{equation} where $\Omega$ is a bounded domain and $V_n$, $u_n$ satisfy \begin{itemize} \item[(i)] $V_n\geq 0$, \item[(ii)] $\|V_n\|_{L^p(\Omega)}\leq C_1$, $\|e^{u_n}\|_{L^{p'}(\Omega)}\leq C_2$ with $p$ ($10$ near the boundary of $\Omega$. We distinguish these two cases in the proof. Consider first that $h<0$ in a neighborhood of $\partial\Omega$. Since a $\delta$ neighbourhood of the boundary, $ N_{\delta}(\partial\Omega) \subset\Omega^-_{\delta}$ for $\delta>0$ small enough and since solutions to \eqref{Eq1.1} and \eqref{Eq1.2} are uniformly bounded in $\Omega_{\delta}^-$ by the results of section 2, solutions to \eqref{Eq1.1} and \eqref{Eq1.2} are uniformly bounded in a neighborhood of $\partial\Omega$. Now, let us see the more delicate case: $h>0$ near $ \partial\Omega$. We apply different arguments for \eqref{Eq1.1} and \eqref{Eq1.2}. Concerning \eqref{Eq1.1}, we use a blow up analysis as in \cite{ChLi2}. For \eqref{Eq1.2}, we use the moving plane method as in \cite{DeLiNu}. \subsection{Blow up analysis near the boundary for \eqref{Eq1.1}} We will follow the main steps in \cite{ChLi2} to analyse the behaviour of a possible blowing up sequence at boundary $\partial\Omega$. Let $\{\lambda_i,u_i\}$ a sequence of solutions to \eqref{Eq1.1}. If $\{u_i\}$ is not bounded, then from Proposition \ref{P1}, we can assert the existence of the sets \[ S_i:= \left\{x : x \text{ is a local maximum of } u_i \right \} \] and for every $C$, define $\tilde{S}_i(C)$, a subset of $S_i$, consisting of points satisfying: $u_i(x)\geq C$ and for any two points $p,q\in \tilde{S}_i(C)$, \[ u_i(p)d(p,q)^{\frac{n-2}{2}}\geq C. \] Furthermore, we have \[ u_i(x)d(x,\,\tilde{S}_i(C))^{\frac{n-2}{2}}\leq K(C). \] for a constant $K(C)$ depending on $C$. We will show that for $C$ large enough, \begin{equation}\label{bd1} N_{\eta}(\partial\Omega)\cap \tilde{S}_i(C)=\emptyset, \end{equation} where $N_{\eta} (\partial\Omega)$ is a $\eta$-neighborhood of $\partial\Omega$. This will complete the proof in case of \eqref{Eq1.1}. The proof is carried out as follows: \begin{itemize} \item[Step 1] We show that $\tilde{S}_i(C)$ is discrete. \item[Step 2] $N_{\eta}(\partial\Omega)\cap \tilde{S}_i(C)=\emptyset$ for $\eta >0$ small enough and $C$ large enough. \end{itemize} {\it Step 1:} Let us define for a fixed constant $C$, $p_i$, $q_i$ $\in\tilde{S}_i(C)$ such that (thanks to Proposition \ref{P1}) \begin{gather*} \sigma_i:= d(p_i,q_i)=\inf_{p,q\in\tilde{S}_i(C)}d(p,q)\\ d_i := d(p_i,\partial\Omega)\leq d(q_i,\partial\Omega)\\ m_i := u(p_i)^{-\frac{2}{n-2}}. \end{gather*} Arguing by contradiction, we assume that $\sigma_i$ ,$m_i\to 0$ and $p_i $ tend to $p_0 $, a point on the boundary, when $i\to +\infty$. As in \cite{ChLi2}, we distinguish three main cases: \begin{itemize} \item[(1)] $m_i=o(d_i)$ and $\sigma_i=O(d_i)$, \item[(2)] $m_i=o(d_i)$ and $d_i=o(\sigma_i)$, \item[(3)] $d_i=O(m_i)$. \end{itemize} In case 1, we use the rescaling function \[ v_i(x):=\sigma_i^{\frac{n-2}{2}}u_i(\sigma_ix+p_i). \] By definition of $\sigma_i$, $v_i$ has only isolated blow-up points. Moreover, since $\sigma_i=o(d_i)$, we can argue similarly as in subsection 3.1 (see in particular Proposition \ref{nbp}) to get that $\{v_i\}$ is bounded. For this, note that if the sequence $\{v_i\}$ is not bounded, then there exists at least two blow up points with finite distance between them. In case 2, we use the rescaling function \[ v_i(x):=d_i^{\frac{n-2}{2}}u_i(d_ix+p_i). \] Since $d_i=o(\sigma_i)$ and $m_i=o(d_i)$, we are in the situation of one isolated and simple blow up point in a half space (because $D_i=\frac{\Omega-p_i}{d_i}\to H:=\{x,x_n\geq -1\}$ up to some standard geometric transformations). So we cannot use directly the blow up analysis from Section 3.1. Since the proof is similar to section 2.2 in \cite{ChLi2}, we just sketch the proof: From \[ -\Delta v_i=\lambda_id_i^2v_i+\tilde{h}v_i^{\frac{n+2}{n-2}} \] where $\tilde{h}(x)=h(d_ix+p_i)$, we prove that \begin{equation}\label{cvh} v_i(x)v_i(0)\to\, w(x):=a\Big(\frac{1}{|x|^{n-2}} -\frac{1}{|x+e|^{n-2}}+\frac{x_n+1}{2^{n-1}}\Big) \end{equation} with $a=\lim\big[\frac{n(n-2)}{h(p_i)}\big]$ and $e=(0,\dots ,2)$. The contradiction is based on Lemma \ref{poh} (Pohozaev identity) in $D_i\cap B_R$ where $R$ is large. Indeed, multiplying this pohozaev identity by $v_i(0)^2$, we have \begin{align*} &v_i(0)^2\int_{\partial (B_R\cap D_i)} B (R, x, v_i,\nabla v_i)\\ &= v_i(0)^2 \Big\{ \lambda_id_i^2 \int_{B_R\cap D_i}|v_i|^2 -\frac{\lambda_i}{2} d_i^2\int_{\partial (B_R\cap D_i)} x.\nu v_i^2 \Big\}\\ &\quad + v_i(0)^2d_i\frac{n-2}{2n} \int_{B_R\cap D_i} (x \cdot \nabla h) v^{\frac{2n}{n-2}}\\ &\quad -v_i(0)^2 \frac{n-2}{2n} \int_{\partial (B_R\cap D_i)}x.\nu h (d_ix+p_i) v_i^{\frac{2n}{n-2}}, \end{align*} where $\nu$ is the unit outward normal. Now, when $i\to +\infty$, we have \begin{equation} \label{IR} v_i(0)^2\int_{\partial (B_R\cap D_i)} B (R, x, v_i,\nabla v_i) \to I_R=\int_{\partial (B_R\cap H)} B (R, x, w,\nabla w), \end{equation} Moreover, $I_R$ tends to $-\infty$ when $R\to +\infty$ (see \cite{ChLi2} p.76). Furthermore, \begin{equation}\label{IL2} v_i(0)^2\frac{\lambda_i}{2} d_i^2\int_{\partial (B_R\cap D_i)} v_i^2 x.\nu \to 0 \end{equation} which follows from (\ref{cvh}) for any $R$ fixed. Now we claim that \begin{equation}\label{IL} v_i(0)^2d_i\frac{n-2}{2n} \int_{B_R\cap D_i} (x \cdot \nabla h) v^{\frac{2n}{n-2}}\to 0. \end{equation} We need to consider 2 cases: $\nabla h(p_0)\neq 0$ or $\nabla h(p_0)= 0$. Suppose that $p_i\to p_0$ satisfying $\nabla h(p_0)\neq 0$. First, \begin{equation}\label{poh2} \begin{aligned} &d_i\int_{B_{\frac{1}{2}}}\nabla h(d_ix+p_i)v_i^{\frac{2n}{n-2}}\\ &=\frac{2n}{n-2}\int_{\partial B_{\frac{1}{2}}}(\frac{\partial v_i}{\partial \nu}\nabla v_i- \frac{1}{2}|\nabla v_i|^2) \nu +\frac{2n}{n-2} \int_{\partial B_{\frac{1}{2}}}\frac{\lambda_i}{2}v_i^2\nu +\int_{\partial B_{\frac{1}{2}}}hv_i^{\frac{2n}{n-2}}\nu. \end{aligned} \end{equation} Multiplying both sides of (\ref{poh2}) by $v_i(0)^2$ and using (\ref{cest}) to estimate $\lambda_i $ terms, we get \[ d_iv_i(0)^2\leq C. \] from which (\ref{IL}) follows. If $\nabla h(p_0)=0$, we argue as in \cite{ChLi2} (see p. 75).(\ref{IL}) now follows. From (\ref{IR}), (\ref{IL2}), (\ref{IL}) and $\lambda_i\geq 0$ we get the contradiction in {\it case 2}. Finally, in case 3, we define \[ v_i(x):=m_i^{\frac{n-2}{2}}u_i(m_ix+p_i). \] Using Proposition \ref{P1}, we see that $\{v_i\}$ is bounded in $B_r(0)$ with $r$ small enough. Then, using results from subsection 3.1 we can show that \begin{itemize} \item[(i)] either $\{v_i\}$ is weakly convergent to $v_0$ satisfying \begin{equation}\label{vo} \begin{gathered} -\Delta v_0=h(p_0)v_0^{\frac{n+2}{n-2}} \\ v_0(0)=1\quad v_0\big|_{\partial H}\equiv 0, \end{gathered} \end{equation} where $H:=\{x:x_n\geq -b\}$ for some $b>0$ (since $d_i=O(m_i)$), \item[(ii)] or there exists $\tilde{z}_i\in D_i$ such that $d(\tilde{z}_i, \partial D_i)\to 0$ and $v_i(\tilde{z}_i)\to +\infty$ when $i\to +\infty$. \end{itemize} In the first case, we get the contradiction since there is no solution to (\ref{vo}). In the second case, let $z_i$ the preimage of $\tilde{z}_i$ and $r_i=d(z_i,\partial\Omega)$. Using the definition of $\sigma_i$, $w_i(x):=u_i(r_ix+p_i)$ has only one possible blow up point in a half space. In this case, we argue as in {\it case 2} to get the contradiction. If $\{w_i\}$ is bounded, we use again (\ref{vo}) to get the contradiction. Thus, we have proved that $\tilde{S}_i$ is discrete in a neighborhood of $\partial\Omega$ for $C$ large. \\ Step 2: To show that $\tilde{S}_i\cap N_{\eta} (\partial\Omega) = \emptyset$, we have just to repeat the arguments in case 2 and case 3. This completes the proof of a priori bound in a neighborhood of $\partial\Omega$ for \eqref{Eq1.1}. \subsection{A priori bound in a neighborhood of $\partial\Omega$ for \eqref{Eq1.2}} Here we deal with \eqref{Eq1.2}. We get the a priori bound using the moving plane method which implies that solutions to \eqref{Eq1.2} are nonincreasing near the boundary. Precisely, we proceed as in \cite{DeLiNu}. Let $(\lambda, u)$ be a solution to \eqref{Eq1.2}. We prove \begin{itemize} \item[(1)] First an integral estimate: \begin{equation}\label{eest} \int_{\Omega}e^u\phi_1\leq C \end{equation} where $\phi_1$ is the first eigenfunction of $-\Delta$ with Dirichlet boundary conditions satisfying $\|\phi_1\|_{\infty}=1$. \item[(2)] For any $x_0$ in $\partial\Omega$, \begin{equation}\label{eeest} u\leq C\quad\text{in }B_{\delta}(x_0)\cap\Omega \end{equation} where $C$ and $\delta$ do not depend on $x_0$. \end{itemize} \noindent\emph{Proof of \eqref{eest}:} Multiplying \eqref{Eq1.2} by $\phi_1$ and integrating by parts, we get \begin{equation}\label{ie1} \lambda_1\int_{\Omega}u\phi_1=\lambda\int_{\Omega}u\phi_1 +\int_{\Omega}h(x)\phi(u)e^u\phi_1. \end{equation} Using that $\{x: h(x)\geq\delta\}=\Omega^+_{\delta}\supset N_{\epsilon} (\partial\Omega)$, where $N_{\epsilon}(\partial\Omega)$ is a $\epsilon$-neighborhood of $\partial\Omega$, and that \begin{equation}\label{ie3} u(x)\leq C \quad\text{ in }\,\Omega\backslash(N_{\epsilon} (\partial\Omega)) \end{equation} with $C$ not depending on $u$, we have \begin{equation}\label{ie2} \lambda\int_{\Omega}u\phi_1+\int_{\Omega}h(x)\phi(u)e^u\phi_1 \geq(\lambda_1(\Omega)+a)\int_{\Omega}u\phi_1-C_0 \end{equation} where $C_0$ and $a>0$ do not depend on $u$. From (\ref{ie1}) and (\ref{ie2}), we get $\int_{\Omega}u\phi_1\leq C$ and using in addition (\ref{ie3}), we get (\ref{eest}). \noindent\emph{Proof of \eqref{eeest}:} This is done by moving plane method as in section 3 for points in $\Gamma_{\delta}$. We give the details here as some arguments are different. To apply moving plane method in a neighbourhood of $x_0$, we need that $\partial\Omega\cap B_{\delta}(x_0)$ is convex. If it is not convex, we make a kelvin transform. Precisely, without generality, we can assume $x_0=0$ and the unit outward normal belongs to the $x_1$-axis. Then, taking $y_0$ in the $x_1$-axis such that $B_{\epsilon_0}(y_0)\cap\Omega=\emptyset$ and $\partial B_{\epsilon_0}(y_0)$ tangent in $x_0$ to $\partial\Omega$, we define the inversion $I(y_0)$ defining in $\mathbb{R}^2\backslash\{y_0\}$ as \[ I(y_0): x\to y=y_0+\frac{|y_0|^2}{|x-y_0|^2}(x-y_0) \] and \[ v(x)=u(y_0+|y_0|^2\frac{(x-y_0)}{|x-y_0|^2})\quad \text{for $x$ in }\Omega^*=I(y_0)^{-1}(\Omega)\subset B_{\epsilon_0}(y_0) \] which satisfies \[ -\Delta v=\frac{|y_0|^4}{|x-y_0|^4}(\lambda v+h(y_0+|y_0|^2\frac{(x-y_0)}{|x-y_0|^2})\phi(v)e^v). \] Now, we define \[ f(x,v)=\frac{|y_0|^4}{|x-y_0|^4}(\lambda v+h(y_0+|y_0|^2\frac{(x-y_0)}{|x-y_0|^2})\phi(v)e^v). \] Therefore, since $v=0$ on $I(y_0)^{-1}(\partial\Omega)$ which is strictly convex near $x_0$, the moving plane method can be carried out if \begin{equation}\label{mvp} \frac{\partial f(x,v)}{\partial \nu}\leq 0 \end{equation} Let \[ \Sigma_{\epsilon}=\{x\in\Omega^*\,\mid\mathop{\rm dist}(x,\partial\Omega^*) \leq\epsilon\}. \] Then, we prove (\ref{mvp}) in $\Sigma_{\epsilon}\cap B_{\epsilon_1}(x_0)$ as follows: \begin{equation}\label{mvp2} \begin{aligned} \frac{\partial f(x,v)}{\partial \nu} &=|y_0|^4(\lambda v+h\phi(v)e^v)\partial_\nu\frac{1}{|x-y_0|^4} \\ &\quad +\frac{|y_0|^4}{|x-y_0|^4}\partial_\nu h(y_0+|y_0|^2\frac{(x-y_0)}{|x-y_0|^2})\phi(v)e^v \\ &\leq -\frac{C_1}{|x-y_0|^5}|y_0|^4(h\phi(v)e^v) +\frac{C_2\|\nabla h\|_{\infty}}{|x-y_0|^6}|y_0|^6\phi(v)e^v \\ &=\frac{|y_0|^4}{|x-y_0|^5}(-C_1h+C_2\|\nabla h\|_{\infty} \frac{|y_0|^2}{|x-y_0|})\phi(v)e^v \\ &\leq \frac{|y_0|^4}{|x-y_0|^5}(-C_1\delta +C(|I(y_0)x-y_0|)\phi(v)e^v \\ &\leq \frac{|y_0|^4}{|x-y_0|^5}(-C_1\delta +C(|y_0|+\epsilon +\epsilon_1))\phi(v)e^v \leq 0 \end{aligned} \end{equation} since $h\geq \delta >0$, $C_1>0$ and for $|y_0|$, $\epsilon_1$, $\epsilon$ small enough. Now, we proceed as in \cite{DeLiNu} (see also \cite{ChLi3}): the unit outward directions in $x\in B_{\epsilon_1}(x_0)\cap\partial\Omega^*$ forms a cone centered at $(1,0)$ with a positive angle $\theta$. Let \[ I=\{\nu\in\mathbb{R}^2: \nu .(1,0)\geq\,|\nu|\cos\theta,\, |\nu|\leq\frac{1}{2}\epsilon_1\} \] be a piece of cone and $I_x=\{x-\nu\mid\,\nu\in I\}$. Then by above arguments, we have \[ v(y)\geq v(x)\quad \text{for }x\in B_{\frac{\epsilon_1}{2}}(x_0)\cap\Omega^* ,\,y\in I_x\cap\Omega^*. \] Using (\ref{eest}), we get \[ \int_{\Omega^*}f(x,v)\mathop{\rm dist}(x,\partial\Omega^*)dx\leq C. \] Therefore, for $x\in B_{\frac{\epsilon_1}{2}}(x_0)\cap\Omega^*$, \begin{align*} C &\geq \int_{\Omega^*}f(y,v)\mathop{\rm dist}(y,\partial\Omega^*)dy\\ &\geq k\int_{I_x}\phi(v(y))e^{v(y)}\mathop{\rm dist}(y,\partial\Omega^*)dy\\ &\geq k_1\phi(v(x))e^{v(x)} \end{align*} since $\phi(\cdot)e^{\cdot}$ is increasing. Thus, $v(x)\leq C$ for $x\in B_{\frac{\epsilon_1}{2}}(x_0)\cap\Omega^*$ which implies (\ref{eeest}) for some $\delta$ depending on $|y_0|$. The proof is now completed for solutions to \eqref{Eq1.2}. \section{Existence of solutions to \eqref{Eq1.1} and \eqref{Eq1.2}} \begin{proof}[Proof of Theorem \ref{th1} and \ref{th2}] First, note that from previous sections, we have uniform a priori bound of solutions to \eqref{Eq1.1} in $[0,\Lambda]\times L^{\infty}(\Omega)$ with $\Lambda <\lambda_1(\Omega^0)$. From assumption (H4)-3 and bootstrap arguments, we have that any solution $(\lambda,u)$ to \eqref{Eq1.1} such that $0 \leq\lambda\leq\lambda_1(\Omega^+)$ satisfies: \begin{equation}\label{est6} \|u\|_{C^1_0(\Omega)}\leq C. \end{equation} Multiplying \eqref{Eq1.1} by $\phi^1_{\Omega^+}$ and integrating by parts, we get that any non trivial solution $(\lambda,u)$ satisfies $\lambda\leq \lambda_1(\Omega^+)$. Thus, (\ref{est6}) is also true for any nontrivial solution $(\lambda, u)$ with $\lambda\geq 0$. Now, the existence of $\mathcal{C}$ follows from global bifurcation theory of Rabinowitz (see \cite{Ra}). Furthermore, from (\ref{est6}), $\mathcal{C}$ reaches $\{\lambda=0\}$ and $\Pi_{\mathbb{R}}\mathcal{C}=[0,\lambda*]$ with $\lambda*<\lambda_1(\Omega^+)$. This completes the proof of 1-(i) and 2-(i). To prove 1-(ii) and 2-(ii), we have just to prove that the bifurcation from $\lambda_1(\Omega)$ is supercritical (i.e. goes towards the right). It implies that $\lambda*>\lambda_1(\Omega)$. To get the supercritical branching, we use standard arguments: From $\int_{\Omega}h{\phi^1_{\Omega}}^{\frac{2n}{n-2}}<0$ and the main result from \cite{CR}, we show that the unique curve of solutions to \eqref{Eq1.1} emanating from $\lambda_1(\Omega)$ is defined only for $\lambda >\lambda_1(\Omega)$ (see \cite{TOu} for more details). To prove that the solution $(0, u_0)$ is such that $u_0 > 0$, we can follow the arguments from the proof of Theorem 1.2, section 6 of \cite{GiPrRa}. This completes the proof of Theorem \ref{th1} and \ref{th2}. \end{proof} \begin{thebibliography}{99} \bibitem{AdGi} Adimurthi and J. Giacomoni, \emph{Bifurcation problems for superlinear elliptic indefinite equations with exponential growth}, to appear in NoDeA Nonlinear Differential Equations Appl. \bibitem{AmLo} H. Amann and J. Lopez G\'omez, \emph{A priori bounds and multiple solutions for superlinear indefinite elliptic problems}. Journal of Differential Equations, {\bf 146} (1998), 336-374. \bibitem{BeCaNi} H. Berestycki, I. Capuzzo Dolcetta and L. Nirenberg, \emph{Superlinear indefinite elliptic problems and nonlinear Liouville Theorems}. Topological Methods in Nonlinear Analysis, {\bf 4} (1994), 59-78. \bibitem{BiGi} I. Birindelli and J. Giacomoni, \emph{Bifurcation Problems for superlinear elliptic indefinite equations}. Topological Methods in Nonlinear Analysis, {\bf 16} ( 2000), 17-36. \bibitem{BrMe} H. Brezis and F. Merle, \emph{Uniform estimates and blow-up behavior for solutions of $-\Delta u=V(x)e^u$ in two dimensions}. Comm. Partial Diff. Equations {\bf 16} (1991), 1223-1253. \bibitem{BrTu} H. Brezis, R.E.L. Turner, \emph{On a class of superlinear elliptic problems}. Comm. Diff. Eq. {\bf 2} (1977), 601-614. \bibitem{CeGr} K. Cerqueti and M. Grossi, \emph{Local estimates for a semilinear elliptic equation with Sobolev critical exponent and application to a uniqueness result}. NoDEA Nonlinear Differential Equations Appl. {\bf 8} (2001), no.3, 251--283. \bibitem{ChLi} W. Chen and C. Li, \emph{A priori estimates for prescribing scalar curvature equations}. Ann. of Math. {\bf 145} (1997), no.3, 547-564. \bibitem{ChLi2} W. Chen and C. Li, \emph{Indefinite elliptic problems with critical exponents}. Advances in nonlinear partial differential equations and related areas (Beijing, 1997), 67--79, World Sci. Publishing, River Edge, NJ, 1998. 35J65 \bibitem{ChLi1} W. Chen and C. Li, \emph{Indefinite elliptic problems in a domain. Discrete Contin}. Dynam, Systems {\bf 3} (1997), no.3, 333-340. \bibitem{ChLi3} W. Chen and C. Li, \emph{A Priori Estimates for Solutions to Nonlinear Elliptic Equations}. Arch. Rational Mech. Anal. {\bf 122} (1993), 145-157. \bibitem{CR} M. G. Crandall and P. H. Rabinowitz, \emph{Bifurcation, perturbation of simple eigenvalues and linearized stability}. Arch. Rational Mech. Anal. {\bf 52} (1973), 161-180. \bibitem{DeLiNu} D. G. De Figueiredo , P. L. Lions and R. D. Nussbaum., \emph{A priori estimates and existence of positive solutions of semilinear elliptic equations}. J. Math. Pures Appli. {\bf 61} (1982), 41-63. \bibitem{GiPrRa} J. Giacomoni, J. Prajapat and M. Ramaswamy, \emph{Positive solution branch for elliptic problems with critical indefinite nonlinearity}, to appear in Differential and Integral Equations. \bibitem{GiSp} B. Gidas and J. Spruck, \emph{A priori bounds for positive solutions of nonlinear elliptic equations}. Commun. P. D. E., {\bf 6}, 883-901. \bibitem{GT} D. Gilbarg and N. S. Trudinger. \emph{Elliptic Partial differential Equations of Second Order.} Springer-Verlag, New York, 1983. \bibitem{Ra} P. H. Rabinowitz, \emph{Some global results for nonlinear eigenvalue problems}. J. Funct. Anal. {\bf 7} (1971), 487-517. \bibitem{RaTeTr} M. Ramos , S. Terracini, C. Troestler, \emph{Superlinear indefinite elliptic problems and Pohozaev type identities}. J. Funct. Anal., {\bf 159}, 596-628. \bibitem{Yl} Yanyan Li, \emph{Prescribing scalar curvature on $S^n$ and related problems}, Part 1. J. of Differential Equations, Vol. {\bf 120} (1995), no.2, 219-410. \bibitem{yz} Yanyan Li and M.Zhu, \emph{Yamabe type equations on three dimensional Riemannian Manifolds}. Comm. in Contemporary Math., Vol. {\bf 1} (1999), no.1, 1-50. \bibitem{TOu} T. Ouyang, \emph{On the positive Solutions of Semilinear equations $\Delta u + \lambda u + h u^p =0 $ on compact Manifolds}. Part II, Indiana Univ Math Jl, Vol {\bf 40} (1991), no.3, 1083-1141 \bibitem{SC} Schoen and Zhang, \emph{Prescribed scalar curvature on the n-sphere}. Calculus of Variations {\bf 4} (1996), 1-25. \end{thebibliography} \end{document}