\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 96, pp. 1--10.\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} \title[\hfilneg EJDE-2007/96\hfil Positive solutions] {Positive solutions for classes of multiparameter elliptic semipositone problems} \author[S. Caldwell, A. Castro, R. Shivaji, S. Unsurangsie\hfil EJDE-2007/96\hfilneg] {Scott Caldwell, Alfonso Castro,\\ Ratnasingham Shivaji, Sumalee Unsurangsie} % in alphabetical order \address{Scott Caldwell \newline Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA} \email[S. Caldwell]{pscaldwell@yahoo.com} \address{Alfonso Castro \newline Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA} \email{castro@math.hmc.edu} \address{Ratnasingham Shivaji \newline Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA} \email{shivaji@ra.msstate.edu} \address{Sumalee Unsurangsie \newline Mahidol University, Thailand} \thanks{Submitted November 13, 2006. Published June 29, 2007.} \subjclass[2000]{35J20, 35J65} \keywords{Positive solutions; multiparameters; mountain pass lemma; \hfill\break\indent sub-super solutions; semipositone} \begin{abstract} We study positive solutions to multiparameter boundary-value problems of the form \begin{gather*} - \Delta u =\lambda g(u)+\mu f(u)\quad \text{in } \Omega \\ u =0 \quad \text{on } \partial \Omega , \end{gather*} where $\lambda >0$, $\mu >0$, $\Omega \subseteq R^{n}$; $n\geq 2$ is a smooth bounded domain with $\partial \Omega $ in class $C^{2}$ and $\Delta $ is the Laplacian operator. In particular, we assume $g(0)>0$ and superlinear while $f(0)<0$, sublinear, and eventually strictly positive. For fixed $\mu$, we establish existence and multiplicity for $\lambda $ small, and nonexistence for $\lambda $ large. Our proofs are based on variational methods, the Mountain Pass Lemma, and sub-super solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \begin{section}{Introduction} We study the multiparameter elliptic boundary-value problem \begin{equation}\label{bvp} \begin{gathered} -\Delta u =\lambda g(u)+\mu f(u)\quad \text{in } \Omega \\ u =0 \quad \text{on } \partial \Omega , \end{gathered} \end{equation} where $\lambda >0$, $\mu >0$, $\Omega \subseteq R^{n}$; $n\geq 2$ is a smooth bounded domain with $\partial \Omega $ in class $C^{2}$ and $\Delta $ is the Laplacian operator. We assume $g:[0,\infty)\to \mathbb{R} $ is differentiable, $g(0)>0$, non decreasing, and there exist $A,B\in (0,\infty )$ and $q\in (1,\tfrac{n+2}{n-2})$ such that for $x>0$ and large \begin{equation}\label{defq} Ax^{q}\leq g(x)\leq Bx^{q}. \end{equation} Also, we assume there exists $\theta >2$ such that for $x>0$ and large \begin{equation}\label{deftheta} xg(x)\geq \theta G(x) \end{equation} where $G(x)=\int_{0}^{x}g(t)dt$. Further, we assume $f:[0,\infty )\to \mathbb{R}$ is differentiable, $f(0)<0$, non decreasing, eventually strictly positive, and there exists $\alpha \in (0,1)$ such that \begin{equation} \label{defalpha} \lim_{u\to \infty }\frac{f(u)}{u^{\alpha }}=0. \end{equation} We establish the following results: \begin{theorem}\label{theo1} Let $\mu >0$ be fixed. There exists $\lambda ^{\ast }>0$ such that if $\lambda \in (0,\lambda ^{\ast })$, {\textup{\eqref{bvp} }} has a positive solution $u_{\lambda }$ satisfying $\|u_{\lambda }\|_{\infty }\geq c^*\lambda ^{-\frac{1}{q-1}}$, where $c^*>0$ is independent of $\lambda $. \end{theorem} \begin{theorem}\label{theo2} There exists $\mu _{0}>0$ such that for $\mu \geq \mu_{0}$, \eqref{bvp} has at least two positive solutions for $\lambda $ small. \end{theorem} \begin{theorem}\label{theo3} Let $\mu >0$ be fixed. Then \eqref{bvp} has no positive solution for $\lambda $ large. \end{theorem} We note that for fixed $\mu >0$, when $\lambda $ is small $\lambda g(0)+\mu f(0)<0$, and hence \eqref{bvp} is a semipositone problem. It has been well documented in recent years (see \cite{kb-rs,ac-cm-rs,ac-rs1}), that the study of positive solutions for semipositone problems is mathematically very challenging. We establish Theorem \ref{theo1} using the Mountain Pass Lemma. In Theorem \ref{theo2}, the second positive solution is established via sub-super solutions. The nonexistence result in Theorem \ref{theo3} is proved by using the fact that $\lambda g(u)+\mu f(u)$ is bounded below by a piecewise linear function. We will prove Theorem \ref{theo1} in Section 2, Theorem \ref{theo2} in Section 3, and Theorem \ref{theo3} in Section 3. Our results apply, for example, to the case when $f(u)=(u+1)^{\frac{1}{3}}-2$ and $g(u)=u^{3}+1$. We refer the reader to \cite{sc-rs-jz} where the case $n=1$ was studied in detail. In particular, using a modified quadrature method, analysis of positive solution curves and their evolution as $\lambda ,\mu $ vary was established. See \cite{su} for related results for single parameter semipositone problems. \end{section} \begin{section}{Proof of Theorem \ref{theo1}} We extend $g$ and $f$ as $g(x)=g(0)$ and $f(x)=f(0)$ for all $x<0$. Throughout this paper we will denote by $W$ the Sobolev space $W_{0}^{1,2}(\Omega )$ and by $L^{r}$ the space $L^r(\Omega)$, for $r \in [1, \infty)$. Let $J: W \to \mathbb{R}$ be defined by \begin{equation}\label{defJ} J(u):=\int_{\Omega }\frac{|\nabla u|^{2}}{2}dx -\int_{\Omega }H_{\lambda }(u)dx, \end{equation} where $H_{\lambda }(u)=\lambda G(u)+\mu F(u)$ with $G(t)=\int_0^tg(s)ds$ and $F(t)=\int_{0}^{t}$ $f(s)ds$. For future reference we note that there exist real numbers $\tilde A, \tilde B, \tilde C$ such that \begin{equation}\label{propG} \begin{gathered} G(x) \leq B\frac{|x|^{q+1}}{q+1} + \tilde B \quad \hbox{for all } x \in \mathbb{R}, \\ G(x) \geq A\frac{x^{q+1}}{q+1} + \tilde A \quad \hbox{for all } x \in [0, \infty), \\ F(x) \leq |x|^{\alpha+1} + \tilde C \quad \hbox{for all } x \in \mathbb{R}. \end{gathered} \end{equation} In addition, defining $h_{\lambda}(x) = \lambda g(x) + \mu f(x)$ it follows from \eqref{defq} that for any $\theta_1 \in (2, \theta)$, there exists $\theta_2$ such that \begin{equation}\label{theta1} x h_{\lambda}(x) \geq \theta _1 (\lambda G(x) + \mu F(x) - \theta_2) \quad \hbox{for all } x \in \mathbb{R}. \end{equation} Also from (\ref{defq}) and (\ref{defalpha}) we see that there exists $\theta_3$ such that \begin{equation}\label{theta3} \begin{gathered} | g (x)| \leq \theta _3( |x|^q + 1) \quad \hbox{for all } x \in \mathbb{R}. \\ | f (x)| \leq \theta _3( |x| + 1) \quad \hbox{for all } x \in \mathbb{R}. \end{gathered} \end{equation} It is well known that $J$ is class $C^1$ and that $u$ is a critical point of $J$ if and only if $u$ is a solution of \eqref{bvp}. We prove $J$ has a critical point using the Mountain Pass Lemma (see Ambrosetti and Rabinowitz in \cite{aa-pr}). We now recall the Mountain Pass Lemma. \begin{lemma}[Mountain Pass Lemma] Let $E$ be a real Banach space and $J\in C^{1}(E,\mathbb{R})$ satisfy the Palais-Smale condition. Suppose $J(0)=0$ and \begin{itemize} \item[(I)] there are constants $\rho ,\alpha >0$ such that $J/_{\partial B_{\rho }}\geq \alpha $ and \item[(II)] there is an $e\in E\backslash \overline{B_{\rho }}$ such that $J(e)\leq 0$. \end{itemize} Then $J$ possesses a critical value $c_0\geq \alpha $. Moreover, $c_0$ can be characterized as \begin{equation*} c_0=\inf_{\sigma \in \Gamma }\max_{t\in \sigma [(0,1)]}J(t), \end{equation*} where $\Gamma =\{ \sigma \in C([0,1] ,E): \sigma (0)=0,\sigma (1)=e \} $ and $B_{\rho }$ is a ball in $E$ with center $0$ and radius $\rho $. \end{lemma} We recall that $J:W \to \mathbb{R}$ is said to satisfy the Palais-Smale condition if every sequence $(v_{n})$, such that $(J(v_{n}))$ is bounded and $\nabla J(v_{n})\to 0$, has a convergent subsequence. Due to (\ref{theta1}) a standard argument (see \cite{aa-pr}) shows that for each $\lambda > 0$, the functional $J$ satisfies the Palais-Smale condition. In Lemma \ref{mp} we show that $J$ satisfies the first and second conditions of the Mountain Pass Lemma and obtain a critical estimate on $J$. In Lemma \ref{regul} we obtain a crucial regularity estimate which we will use to prove that the solution obtained from the Mountain Pass Lemma is positive. In the next lemma we prove that $J$ satisfies the remaining conditions of the Mountain Pass Lemma and obtain an estimate on the critical level. \begin{lemma}\label{mp} There exists $\overline{\lambda }>0$ and $C>0$ such that if $\lambda \in (0,\overline{\lambda })$ then $J$ has a critical point $u_{\lambda}$ of mountain pass type satisfying \begin{equation*} J(u_{\lambda })\geq \frac{C^2}{8} \lambda ^{-\tfrac{2}{q-1}}. \end{equation*} \end{lemma} \begin{proof} By the Sobolev imbedding theorem there exist positive constants $K_1, K_2$ such that \begin{equation}\label{sobolev} \|u\|_{L^{q+1}(\Omega)} \leq K_1 \|u\|_{W^{1,2}_0(\Omega)}, \quad \hbox{and}\quad \|u\|_{L^{\alpha +1}(\Omega)} \leq K_2 \|u\|_{W^{1,2}_0(\Omega)}, \end{equation} for all $u \in W^{1,2}_0(\Omega)$. Let $C = ((q+1)/(4BK_1^{q+1}))^{1/(q+1)}$ and $r= C\lambda ^{-\tfrac{1}{q-1}}$. Let $\|u\|_{W_{0}^{1,2}}=r$. This and (\ref{propG}) yield \begin{equation}\label{lemmp1} \begin{aligned} J(u)& =\frac{1}{2}\ r^{2}-\int_{\Omega}H_{\lambda }(u)dx \\ & \geq \frac{1}{2}\ r^{2}-\frac{\lambda B}{q+1} \int_{\Omega }|u|^{q+1} dx - \lambda \tilde B|\Omega| - \mu \int_{\Omega }|u|^{\alphaÊ+1 } dx- \mu \tilde C |\Omega| \\ & \geq \frac{1}{2}\ r^{2}-\frac{\lambda BK_1^{q+1}}{q+1}r^{q+1} - \lambda \tilde B|\Omega| - \mu K_2^{\alpha +1} r^{\alpha +1} - \mu \tilde C |\Omega| \\ & = \lambda ^{-2/(q-1)}\Big(\frac{C^2}{4} - \lambda ^{(q+1)/(q-1)} \tilde B|\Omega| - \mu K_2^{\alpha +1} C^{\alpha +1}\lambda^{(1-\alpha)/(q-1)} \\ & \quad - \mu \tilde C |\Omega| \lambda^{2/(q-1)}\Big)\\ & \geq \lambda ^{-2/(q-1)}\frac{C^2}{8} \end{aligned} \end{equation} for $\lambda$ sufficiently small. Let $v_{1}$ denote an eigenfunction corresponding to the principal eigenvalue $\lambda _{1}$ of $-\Delta $ with Dirichlet boundary conditions with $v_{1}>0$ and $\|v_{1}\|_{W_{0}^{1,2}}=1$. Let \begin{equation}\label{defbeta} F(\beta) = \min\{F(s); s \in [0,\infty)\}. \end{equation} For $s \geq 0$ \begin{equation}\label{lemmp3} \begin{aligned} J(sv_{1})&= \frac{s^{2}}{2}\|v_{1}\|_{W_{0}^{1,2}( \Omega ) }^2-\lambda \int_{\Omega }G(sv_{1})dx-\mu \int_{\Omega }F(sv_{1})dx \\ & \leq \frac{s^{2}}{2}- \lambda \Big(As^{q+1} \int_{\Omega } \frac{v_{1}^{q+1}}{q+1} dx+ \tilde A |\Omega| \Big)-\mu F(\beta )|\Omega |\\ &\to -\infty \text{ as }s\to \infty, \end{aligned} \end{equation} since $q>1$. This implies there is a $s_{1}>r$ such that $J( s_{1}v_{1})\leq 0$. By choosing $v=s_{1}v_{1}$ we have satisfied the second condition of the Mountain Pass Lemma and Lemma \ref{mp} is proven. \end{proof} \begin{lemma}\label{regul} There exist $c_1>0$ and $\hat \lambda \in (0, \bar \lambda)$, such that $\|u_{\lambda }\|_{\infty }\leq c_{1}\lambda ^{\frac{-1}{q-1}}$ for all $\lambda \in (0, \hat \lambda)$. \end{lemma} \begin{proof} Throughout this proof $c$ denotes several positive constants independent of the parameter $\lambda$. From (\ref{propG}) we have \begin{equation}\label{regul1} \begin{aligned} J(s v_1) & =\frac{1}{2}\ s^{2}-\int_{\Omega}H_{\lambda }(sv_1)dx \\ & \leq \frac{1}{2}\ s^{2}-\frac{\lambda As^{q+1} }{q+1} \int_{\Omega }|v_1|^{q+1} dx - \lambda \tilde A|\Omega| - \mu F(\beta) |\Omega| \\ & \leq \frac{1}{2}\ s^{2}-\frac{\lambda AK_2}{q+1}s^{q+1} - (\mu F(\beta) + \lambda \tilde A) |\Omega| \mbox{ where $K_2=\int_\Omega |v_1|^{q+1}dx$}\\ & \equiv p(s) - (\mu F(\beta)+ \lambda \tilde A) |\Omega|. \end{aligned} \end{equation} Since \begin{equation} p(s) \leq \Big(\frac{1}{2} - \frac{1}{q+1}\Big) (AK_2)^{-2/(q-1)}\lambda^{-2/(q-1)} \end{equation} for $s \in [0, \infty)$, there exists a positive constant $c$ such that for $ \lambda>0$ sufficiently small \begin{equation} J(sv_1) \leq c\lambda^{-2/(q-1)} \quad \hbox{for all } s \in [0, \infty). \end{equation} Since $J(u_{\lambda}) \leq \max\{J(sv_1); s \in [0, s_1]\}$ we have \begin{equation} J(u_{\lambda}) \leq c \lambda^{-2/(q-1)}, \end{equation} for $\lambda>0$ sufficiently small. From (\ref{theta1}), for $\lambda $ small we have \begin{equation}\label{regul2} \begin{aligned} \|u\|_{W^{1,2}_0(\Omega) }^2 &\leq 2 c\lambda^{-2/(q-1)} +2 \int_{\Omega }H_{\lambda}(u_{\lambda} ) dx \\ & \leq 2 c\lambda^{-2/(q-1)} +\frac{2}{\theta_1} \int_{\Omega }u_{\lambda}h_{\lambda}(u_{\lambda} ) dx + 2\theta_2|\Omega| \\ & = 2 c\lambda^{-2/(q-1)} +\frac{2}{\theta_1} \|u\|_{W^{1,2}_0(\Omega) }^2 + 2\theta_2|\Omega| . \end{aligned} \end{equation} Since $\theta_1 > 2$, from (\ref{regul2}) we see that there exists $c>0$ such that for $\lambda$ small \begin{equation}\label{regul3} \|u_{\lambda}\|_{W^{1,2}_0(\Omega) } \leq c\lambda^{-1/(q-1)}. \end{equation} This, (\ref{theta1}), and the fact that $u_{\lambda}$ is a critical point of $J$ also give \begin{equation}\label{regul4} \int_{\Omega }u_{\lambda}h_{\lambda}(u_{\lambda} ) dx \leq c\lambda^{-2/(q-1)} \quad \hbox{and}\quad \int_{\Omega }H_{\lambda}(u_{\lambda} ) dx \leq c\lambda^{-2/(q-1)} . \end{equation} From (\ref{regul3}) and the Sobolev imbedding theorem, for $\lambda>0$ small, $\|u_{\lambda}\|_{L^{2n/(n-2)} }\leq Kc\lambda^{-1/(q-1)}$ where $K>0$ is the positive constant given in this imbedding. Hence using (\ref{theta3}) and letting $a_1 = |\Omega|^{\frac{(q-1)(n-2)}{2n}}$, $a_2 =|\Omega| ^{\frac{q(n-2)}{(2n)}} $ we have \begin{equation}\label{regul5} \begin{aligned} \|h_{\lambda}(u_{\lambda})\| _{L^{2^*/q} } & \leq \theta_3\Big(\int_{\Omega } ({\lambda} |u_{\lambda}|^{q} + \mu |u_{\lambda}| + (\lambda+\mu))^ {\frac{2n}{(q(n-2))}} dx\Big)^{\frac{q(n-2)}{(2n)}} \\ & \leq \theta_3\left(\lambda \|u_{\lambda}\|_{L^{2^* }}^q + \mu a_1\|u_{\lambda}\|_{L^{2^*}} + (\lambda + \mu)a_2 \right)\\ & \leq \theta_3\left(\lambda K^q \|u_{\lambda}\|_{W }^q + \mu a_1 K \|u_{\lambda}\|_{W } + (\lambda + \mu)a_2 \right),\\ \end{aligned} \end{equation} Since the constants $\theta_3, K, \mu, a_1, a_2$ in (\ref{regul5}) are independent of $\lambda$, from (\ref{regul3}) we see that there exists a positive constant $c$ such that for $\lambda $ small enough \begin{equation}\label{regul6} \|h_{\lambda}(u_{\lambda})\| _{L^{2^*/q} } \leq c\lambda^{-1/(q-1)} . \end{equation} By a priori estimates for elliptic boundary-value problems (see \cite{sa-ld-ln}) $\|u_{\lambda}\|_2 \leq c \lambda^{-1/(q-1)}$, where $\| \ \|_2$ denotes the norm in the Sobolev space $W^{2,2}(\Omega)$ and $c$ is a constant independent of $\lambda$. Since $W^{2,2}(\Omega)$ may be imbedded into $L^{2n/(n-4)}$ repeating the argument in (\ref{regul5}) and (\ref{regul6}) we see that \begin{equation}\label{regul7} \|h_{\lambda}(u_{\lambda})\| _{L^{2n/(q(n-4))} } \leq c\lambda^{-1/(q-1)} \quad \hbox{and} \quad \|u_{\lambda}\|_{{2}, \frac{ 2n}{q (n-2)}} \leq c \lambda^{-1/(q-1)}, \end{equation} where $ \| \cdot \|_{{2}, \frac{ 2n}{q (n-2)}} $ denotes the norm in the Sobolev space $W^{ {2}, \frac{ 2n}{q (n-2)}}(\Omega)$. Iterating this argument we conclude that \begin{equation}\label{regul8} \|u_{\lambda}\|_{{2}, r } \leq c \lambda^{-1/(q-1)}, \end{equation} with $r > n/2$. Since for such $r's$, $W^{2,r}$ is continuously imbedded in $L^{\infty}$, we have $\|u_{\lambda}\| \leq c \lambda^{-1/(q-1)}$, which proves the lemma. \end{proof} \begin{proof}[Proof of Theorem \ref{theo1}] From the definition of $g$ we see that $G$ is bounded from below. We let $\hat G = \inf\{G(s); s \in \mathbb{R}\}$. This, Lemma \ref{mp}, and (\ref{defbeta}) give \begin{equation}\label{esthbelow} \begin{aligned} \int_{\Omega }\ h_{\lambda }(u_{\lambda })u_{\lambda} dx & = \|u_{\lambda}\|_W^2 \\ & \geq 2J(u_{\lambda}) + 2(\hat G + F(\beta))|\Omega| \\ & \geq \frac{C^2}{4} \lambda^{-2/(q-1)} + 2(\hat G + F(\beta))|\Omega| \\ & \geq \frac{C^2}{8} \lambda^{-2/(q-1)}, \end{aligned} \end{equation} for $\lambda>0$ small. Let $\gamma > 0$ be such that $|\Omega|\theta_3 \gamma [(\gamma^q + \gamma \mu) = C^2/(32|\Omega|)$ with $C$ as in \eqref{esthbelow}, and $\Omega_{\lambda} = \{x; u_{\lambda}(x) \geq \gamma \lambda^{-1/(q-1)}\}$. From Lemma \ref{regul}, \eqref{esthbelow}, and \eqref{theta3} we have \begin{equation} \begin{aligned} \frac{C^2}{8} \lambda^{-2/(q-1)} & \leq \int_{\Omega }\ h_{\lambda }(u_{\lambda })u_{\lambda} dx \\ & = \int_{\Omega_{\lambda}}\ h_{\lambda }(u_{\lambda })u_{\lambda} dx + \int_{\Omega - \Omega_{\lambda}}\ h_{\lambda }(u_{\lambda })u_{\lambda} dx \\ & \leq |\Omega_{\lambda}| \theta_3 c_1\lambda^{-1/(q-1)}[(c_1^q + c_1 \mu) \lambda^{-1/(q-1)} + \lambda + \mu ] \\ & \quad + |\Omega|\theta_3 \gamma \lambda^{-1/(q-1)}[(\gamma^q + \gamma \mu) \lambda^{-1/(q-1)} + \lambda + \mu ] \\ & \leq 2\theta_3 \lambda^{-2/(q-1)}(|\Omega_{\lambda}| c_1(c_1^q + c_1 \mu) + |\Omega| \gamma (\gamma^q + \gamma \mu) ), \end{aligned} \end{equation} for $\lambda>0$ small. Now by the definition of $\gamma$ we conclude \begin{equation} |\Omega_{\lambda}| \geq \frac{C^2}{32\theta_3c_1(c_1^q + c_1 \mu) } \equiv k_1. \end{equation} Let $z:\bar \Omega \to \mathbb{R}$ be the solution to \begin{equation}\label{defz} \begin{gathered} -\Delta z =1 \quad \hbox{in } \Omega \\ z =0 \quad \hbox{on } \partial \Omega \end{gathered} \end{equation} Since $\Omega$ is assumed to be of class $C^2$, from regularity theory for elliptic boundary-value problems it is well know (see \cite{dg-nt}) that there exist a positive constants $\sigma_1, \sigma_2 $ such that \begin{equation}\label{zc2} \sigma_1 d(x, \partial \Omega) \leq z(x) \leq \sigma_2 d(x, \partial \Omega), \end{equation} where $d(x, \partial \Omega)$ denotes the distance from $x$ to the boundary of $\Omega$. Let $\eta (x)$ denote the inward unit normal to $\Omega $ at $x\in \partial \Omega $. Since $\Omega $ is a smooth region, there exist an $\varepsilon >0$ such that \begin{equation*} N_{\varepsilon }(\partial \Omega )=\left\{ x+\beta \eta ( x):\beta \in [0,\varepsilon ),x\in \partial \Omega \right\} \end{equation*} is an open neighborhood of $\partial \Omega $ relative to $\overline{\Omega }$. Also (see \cite{vg-ap}), this $\varepsilon $ can be chosen small enough so that if $y=x+\beta \eta (x)$ then $d(y,\partial \Omega )=|\beta |$. Since $|N_{\varepsilon }(\partial \Omega )| =O(\varepsilon )\to 0$ as $\varepsilon \to 0$, we can without loss of generality assume that \begin{equation*} |N_{\varepsilon }(\partial \Omega )|\leq \frac{k_1}{2} . \end{equation*} Letting $K_{\lambda}=\Omega_{\lambda} - N_{\varepsilon }(\partial \Omega )$, we have that \begin{equation*} |K_{\lambda}|\geq \frac{k_{1}}{2}. \end{equation*} Let $G$ denote the Green's function of the Laplacian operator, $-\Delta $, in $\Omega $, with Dirichlet boundary condition. For $x\in K_{\lambda}$ and $\xi \in \partial \Omega $ we have, by Hopf's maximum principle, \begin{equation*} \frac{\partial G}{\partial \eta }(x,\xi )>0. \end{equation*} Since $K_{\lambda}\times \partial \Omega $ is compact there exists $\varepsilon _{1}\in (0,\varepsilon )$ and $b>0$ such that if $x\in K_{\lambda}$ and $\xi \in N_{\varepsilon _{1}}(\partial \Omega )$ then \begin{equation*} \frac{\partial G}{\partial \eta }(x,\xi )\geq b. \end{equation*} In particular, for $x\in K_{\lambda}$ and $\ d(\xi ,\partial \Omega ) <\varepsilon _{1}$ we have $G(x,\xi )\geq bd(\xi ,\partial \Omega )$. For $\xi $ such that $d(\xi ,\partial \Omega )<\varepsilon _{1}$ we have \begin{equation*} u_{\lambda }(\xi )=\int_{\Omega }G(x,\xi ) h_{\lambda }(u_{\lambda })dx=\int_{\Omega }G(x,\xi )\lambda g(u_{\lambda })dx+\int_{\Omega }G( x,\xi )\mu f(u_{\lambda })dx. \end{equation*} Since $g( u_{\lambda })>0$ for all $u_{\lambda }$ \begin{equation*} \begin{aligned} u_{\lambda }(\xi )& \geq \int_{K_{\lambda}}G(x,\xi ) \lambda g(u_{\lambda })dx+\int_{\Omega }G(x,\xi )\mu f(u_{\lambda })dx \\ & \geq \int_{K_{\lambda}}G(x,\xi ) \lambda g(u_{\lambda })dx+\mu f(0) z(\xi). \end{aligned} \end{equation*} Therefore, for $\lambda $ small enough by \eqref{defq} and \eqref{zc2}, \begin{equation}\label{defctilde} \begin{aligned} u_{\lambda }(\xi )& \geq \int_{K_{\lambda}}bd(\xi ,\partial \Omega )\lambda \ Au_{\lambda }^{q}dx+\mu f(0) z(\xi) \\ & \geq bd(\xi ,\partial \Omega )A\gamma ^{q}\lambda ^{\frac{-1}{ q-1}}|K_{\lambda}|+\mu f(0)\sigma_2 d(\xi, \partial \Omega) \\ & \geq \tilde c d(\xi, \partial \Omega)\lambda ^{\frac{-1}{ q-1}}, \end{aligned} \end{equation} where $\tilde c>0$ is independent of $\lambda$. We define $w_{\lambda }(x)$ and $z_{\lambda }(x)$ such that \begin{gather*} -\Delta w_{\lambda } =\lambda g(u_{\lambda })+\mu f^{+}( u_{\lambda })\quad\text{in }\Omega \\ w_{\lambda } =0\quad \text{on } \partial \Omega \end{gather*} and \begin{gather*} -\Delta z_{\lambda } =\mu f^{-}(u_{\lambda })\quad\text{in }\Omega \\ z_{\lambda } = 0\quad \text{in }\partial \Omega \end{gather*} where \begin{equation*} f^{+}(x)=\begin{cases} f(x) & x\geq \beta \\ 0 &x<\beta \end{cases} \quad\text{and}\quad f^{-}(x)=\begin{cases} f(x)& x\leq \beta \\ 0& x>\beta\,. \end{cases} \end{equation*} It is clear that $u_{\lambda }=w_{\lambda }+z_{\lambda }$. Also, note that \begin{equation*} z_{\lambda }(x)=\int_{\Omega }G(x,y) \mu f^{-}(u_{\lambda }(y))dy \end{equation*} so clearly $z_{\lambda }\leq 0$ and since $f^{-}(u_{\lambda }(y))$ $\geq f(0)$ we have \begin{equation*} z_{\lambda }(x)\geq \int_{\Omega }G(x,y)\mu f(0)dy=\mu f(0) \int_{\Omega }G(x,y)dy. \end{equation*} So we have $-M_{1}\leq z(x)\leq 0$ where $M_{1}=-\mu f(0)\max_{x\in \overline{\Omega }}\int_{\Omega }G(x,y)dy>0$. For $x $ such that $d(x ,\partial \Omega )=\varepsilon _{1}$ we have \begin{equation*} w_{\lambda }(\xi )=u_{\lambda }(\xi )-z_{\lambda }(\xi )\geq u_{\lambda }(\xi ) \geq \epsilon_1 \tilde c\lambda ^{\frac{-1}{q-1}}, \end{equation*} and by the maximum principle we have $w_{\lambda }(x)\geq \epsilon_1 \tilde c\lambda ^{\frac{-1}{q-1}}$ for all $x\in \Omega - N_{\varepsilon _{1}}(\partial \Omega )$. This implies that $u_{\lambda }(x)=w_{\lambda }(x)+z_{\lambda }( x)\geq \epsilon_1 \tilde c\lambda ^{\frac{-1}{q-1}}-M_{1}$ and so $u_{\lambda }(x)\geq (\epsilon_1 \tilde c/2)\lambda ^{\frac{-1}{q-1}}$ for all $x\in \Omega \backslash N_{\varepsilon _{1}}(\partial \Omega )$ for small $\lambda $. This and \eqref{defctilde} imply that for $\lambda $ small enough $u_{\lambda }(x)>0$ on $\Omega $, which proves Theorem \ref{theo1}. \end{proof} \end{section} \begin{section} {Proof of Theorem \ref{theo2}} In this section we prove a multiplicity result for $\mu >\mu _{0}$ and $\lambda $ small using a sub and super solution method. According to \cite{ac-jg-rs} there exists a $\mu _{0}>0$ such that for $\mu \geq \mu _{0}$ there exists a $w$ such that \begin{gather*} -\Delta w =\mu f(w)\quad \text{in }\Omega \\ w = 0\quad \text{on } \partial \Omega \end{gather*} where $w>0$ on $\Omega $. Since $\lambda >0$ and $g>0$ it follows that \begin{gather*} -\Delta w \leq \lambda g(w)+\mu f(w)\quad \text{in }\Omega \\ w \leq 0\quad \text{on }\partial \Omega , \end{gather*} which implies that $w$ is a sub solution of \eqref{bvp}. Let $z$ be as in \eqref{defz}. Define $\phi =\sigma z$ where $\sigma >0$, independent of $\lambda$, is large enough so $\phi > w$ in $\Omega$ and \begin{equation*} \mu \frac{f(\sigma z)}{\sigma }<\frac{1}{2}. \end{equation*} This is possible since $f$ is a sublinear function (see \eqref{defalpha}). Next let $\lambda >0$ be so small that \begin{equation*} \lambda \frac{g(\sigma z)}{\sigma }<\frac{1}{2}. \end{equation*} Thus \begin{equation*} -\Delta \phi = \sigma \geq \lambda g(\sigma z) + \mu f(\sigma z) = \lambda g(\phi )+\mu f(\phi )\ \quad \text{in }\Omega . \end{equation*} Hence $\phi $ is a supersolution of \eqref{bvp} and there exists a solution $\tilde u_{\lambda}$ (say) of \eqref{bvp} such that $w \leq \tilde u_{\lambda} \leq \phi$ for $\mu \geq \mu_0$ and $\lambda>0$ small. However, from Theorem \ref{theo1}, for $\lambda \,$small, we have the existence of a positive solution, $u_{\lambda }$, such that $\|u_{\lambda }\| _{\infty }\geq c_0\lambda ^{-\frac{1}{q-1}}$. Hence $\lambda $. small $\widetilde{u}_{\lambda }$ and $u_{\lambda }$ are two distinct positive solutions of \eqref{bvp}. \end{section} \begin{section} {Proof of Theorem \ref{theo3}} Let $u$ be a positive solution to \eqref{bvp}. There exist $\sigma >0$ and $\varepsilon >0$ such that $g(u)\geq (\sigma u+\varepsilon )$ for all $u\geq 0$. So for $\lambda >0$, it follows that \begin{equation*} \lambda g(u)+\mu f(u)\geq \begin{cases} \lambda (\sigma u+\varepsilon )&\text{for }u\geq \beta \\ \lambda (\sigma u+\varepsilon )+\mu f(0)&\text{for }u\leq \beta\,. \end{cases} \end{equation*} Choosing $\lambda $ large enough so that $ \lambda \varepsilon +\mu f(0)\geq \frac{\lambda \varepsilon }{2}$, we have \begin{equation*} \lambda g(u)+\mu f(u)\geq \lambda \sigma u +\frac{\lambda \varepsilon }{2} \end{equation*} for $u\geq 0$ and $\lambda $ large. Now let $\lambda _{1}$ be the first eigenvalue and $\phi >0$ be a corresponding eigenfunction of $-\Delta $ with Dirichlet boundary condition. Multiplying both sides of \eqref{bvp} by $\phi $ and integrating we get \begin{equation*} \int_{\Omega }(-\Delta u)\phi dx=\int_{\Omega }(\lambda g(u)+\mu f(u))\phi dx \end{equation*} which implies \begin{gather*} \int_{\Omega }u\lambda _{1}\phi dx=\int_{\Omega }( \lambda g(u)+\mu f(u))\phi dx, \\ \int_{\Omega }u\lambda _{1}\phi dx\geq \int_{\Omega }( \lambda \sigma u+\frac{\lambda \varepsilon }{2})\phi dx, \\ \int_{\Omega }[\lambda _{1}-\lambda \sigma ] u\phi dx\geq \int_{\Omega }\frac{\lambda \varepsilon }{2}\phi dx. \end{gather*} For $\lambda >\frac{\lambda _{1}}{\sigma }$ we obtain a contradiction. So for a given $\mu >0$, \eqref{bvp} has no positive solution for large $\lambda $. \subsection*{Appendix A} (see also \cite{sc} and \cite{su}) Let $10 \end{equation*} since $1\alpha _{0}$. Similarly, \begin{equation*} \alpha _{2}-\alpha _{1}=\alpha _{1}[\frac{n}{qn-2\alpha _{1}}-1] >\alpha _{0}[\frac{n}{qn-2\alpha _{0}}-1] , \end{equation*} so $\alpha _{2}>\alpha _{1}$ and $\alpha _{2}\geq \alpha _{0}+2A( q,n)$. Repeating this argument $p$ times we have $\alpha _{p}\geq \alpha _{0}+pA(q,n)$ and $(\alpha _{j})$ to be increasing in constant increments, which contradicts $2\alpha _{p}