\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Elliptic equations with one-sided critical growth \hfil EJDE--2002/89} {EJDE--2002/89\hfil Marta Calanchi \& Bernhard Ruf \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 89, pp. 1--21. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Elliptic equations with one-sided critical growth % \thanks{ {\em Mathematics Subject Classifications:} 35J20. \hfil\break\indent {\em Key words:} Nonlinear elliptic equation, critical growth, linking structure. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted March 01, 2002. Published October 18, 2002.} } \date{} % \author{Marta Calanchi \& Bernhard Ruf} \maketitle \begin{abstract} We consider elliptic equations in bounded domains $\Omega\subset \mathbb{R}^N $ with nonlinearities which have critical growth at $+\infty$ and linear growth $\lambda$ at $-\infty$, with $\lambda > \lambda_1$, the first eigenvalue of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms provided $N \ge 6$. In dimensions $N = 3,4,5$ an additional lower order growth term has to be added to the nonlinearity, similarly as in the famous result of Brezis-Nirenberg for equations with critical growth. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction} We consider the superlinear problem \begin{equation}\label{p1} \begin{gathered} -\Delta u = \lambda u +u_+^{2^\star -1} +g(x,u_+)+f(x) \quad \mbox{in } \Omega \\ u= 0 \quad \mbox {on } \partial \Omega \end{gathered} \end{equation} where $\Omega \subset \mathbb{R}^N\; (N\ge 3)$ is a bounded domain with smooth boundary, $2^\star =2N/(N-2)$ is the critical Sobolev exponent, $g(\cdot,s_+)\in C (\bar{\Omega}\times \mathbb{R}, \mathbb{R}^+)$ has a subcritical growth at infinity, and $s_+=\max\{s,0\}$. Furthermore, we assume that $\lambda > \lambda_1$, the first eigenvalue of the Laplacian. This means that the function $$ k(s) = \lambda s + s_+^{2^\star -1} + g(x,s_+) $$ ``interferes" with all but a finite number of eigenvalues of the Laplacian, in the sense that $$ \lambda_1 < \lambda = \lim_{s \to - \infty} \frac{k(s)}{s} < \lim_{s \to +\infty} \frac{k(s)}{s} = + \infty $$ For subcritical nonlinearities, such problems have been treated by Ruf-Srikanth \cite{RS} and de Figueiredo \cite{D}, proving the existence of at least two solutions provided that the forcing term $f(x)$ has the form \begin{equation}\label{eq1} f(x) = h(x) + t e_1(x), \end{equation} where $h \in L^r({\Omega})$, for some $r>N$, is given, $e_1$ is the (positive and normalized) first eigenfunction of the Laplacian, and $t > T$, for some sufficiently large number $T=T(h)$. We remark that the search of solutions for forcing terms of the form (\ref{eq1}) is natural if the nonlinearity crosses {\it all} eigenvalues, i.e. $\lambda<\lambda_1$ (then the problem is of so-called Ambrosetti-Prodi type); indeed, in this case there exists an obvious {\it necessary condition} of the form $ \int_\Omega f e_1 dx \lambda_1$\\ ii) $h\in L^r(\Omega), r > N$, is given, with $h\in \ker(-\Delta-\lambda)^\perp$ if $\lambda$ is an eigenvalue.\\ Then there exists $T_0=T_0(h)>0$ such that if $t>T_0$ then problem (\ref{p2a}) has a negative solution $\phi_t\in W^{2,r}\cap W^{1,r}_0\subset C^{1,1-N/r}$. Suppose in addition that\\ iii) $\lambda$ is {\rm not} an eigenvalue of $(-\Delta,H_0^1(\Omega))$ \\ iv) $N \ge 7$ \\ Then problem (\ref{p2a}) has a second solution for $t>T_0$. \end{theorem} We remark that in \cite{DJ} only $h\in L^2(\Omega)$ is assumed, however this seems not sufficient to get the stated results. In this paper we improve and extend the result of de Figueiredo and Jianfu in the following ways: \paragraph{Main Results:} \begin{description} \item[$1)$] $\lambda > \lambda_1$ and $\lambda$ is not an eigenvalue:\\ $N \ge 6$: if $h \in L^r(\Omega)$ ($r > N$), then equation (\ref{p1}) has a second solution for $f = h + te_1$, with $t > T_0(h)$. \\ $N = 3, 4, 5$: if $h \in L^r(\Omega)$ ($r > N$) and $g(x,s_+) \ge c s^q$ for some $q=q(N)>1$ and $c>0$, then equation (\ref{p1}) has a second solution for $f = h + te_1$, with $t > T_0$. \item[$2)$] $\lambda$ is an eigenvalue, i.e. $\lambda = \lambda_k$, for some $k \ge 2$: If $h \in L^r(\Omega)$ ($r > N$) satisfies $h \perp ker (-\Delta-\lambda_k, H_0^1(\Omega))$, then equation (\ref{p1}) has a second solution for $f = h + te_1$ with $t > T_0$ (in dimensions $N = 3,4,5$ the assumptions made in 1) have to be added). \item[$3)$] $\lambda$ in a left neighborhood of $\lambda_k$, $k\ge 2$: There exists a $\delta > 0$ such that if $\lambda \in (\lambda_k - \delta, \lambda_k)$ and $h \in L^r(\Omega)$ ($r > N$) satisfies $h \perp ker (-\Delta, H_0^1(\Omega)$, then equation (\ref{p1}) has at least three solutions for $f = h + te_1$, with $t > T_0$ (in dimensions $N = 3,4,5$ the assumptions made in 1) have to be added). \end{description} For proving these statements, one proceeds as follows: the first (negative) solution is easily obtained (see \cite{RS}, \cite{D}, \cite{DJ}). To obtain a second solution, one uses the saddle point structure around the first solution and applies the generalized mountain pass theorem of Rabinowitz \cite{Ra}. To prove the Palais-Smale condition, one proceeds as in \cite{BN}, using a sequence of concentrating functions (obtained from the so-called Talenti function). However due to the presence of the first solution, lower order terms appear in the estimates. To handle these estimates, an ``orthogonalization" procedure based on separating the supports of the Talenti sequence and the (approximate) first solution is used (this approach was introduced in \cite{GR}). With this method we obtain the results 1) and 2). To prove 3), one shows that the ``branch" of solutions with $\lambda \in (\lambda_k,\lambda_{k+1})$ can be extended to $\lambda = \lambda_k$ if $h \perp ker(-\Delta)$. Actually, this branch can be extended slightly beyond $\lambda_k$, i.e. to $\lambda \in (\lambda_k - \delta, \lambda_k]$, and the corresponding solutions are clearly bounded away from the negative solution. Since in $\lambda_k$ starts a bifurcation branch $(\lambda,u)$ emanating from the negative solution and bending to the left (as shown by de Figueiredo - Jianfu \cite{DJ}), we conclude that for $\lambda$ to the left of and close to $\lambda_k$ there exist at least three solutions. \section{Statement of theorems} In this section we give the precise statements of the theorems. Furthermore, the notation and basic properties are introduced. We consider problem (\ref{p1}) under the following conditions on the nonlinearity $g$: \begin{description} \item[$(g_1)$] $g:\bar{\Omega}\times\mathbb{R}\rightarrow \mathbb{R}^+$ is continuous; \item[$(g_2)$] $g(x,s) \equiv 0$ for $s \le 0$, i.e. $g(x,s) = g(x,s_+)$ with $g(x,0) = 0$; \item[$(g_3)$] There exist constants $c_1>0$ and $10$ and $q$ with $\max \left\{ \frac{N}{2(N-1)} \; ,\; \frac{2}{3} \right\} < \frac{q+1}{2^\star} <1$, such that $g(x,s_+)\ge c(s_+)^{q}$, for all $s \in \mathbb R$. \end{description} We prove the following results: \begin{theorem}\label{t1} $N\ge 6$: Let $h \in L^r(\Omega), r > N$, be given, and let $T_0=T_0(h)$ the number given by Theorem 0. Assume that $\lambda > \lambda_1$, and that $g$ satisfies $(g_1), (g_2), (g_3)$. If $\lambda$ is an eigenvalue, say $\lambda = \lambda_k$, assume in addition that $h \perp ker(-\Delta -\lambda_k)$. Then problem (\ref{p1}) has a second solution for $f = h + te_1$, with $t > T_0$. \end{theorem} \begin{theorem} \label{t2} $3 \le N \le 5$: Let $h \in L^r, r > N$, be given, and suppose that all other assumptions of Theorem \ref{t1} are satisfied. If $g$ satisfies also $(g_4)$, then problem (\ref{p1}) has a second solution for $f = h + te_1$, with $t > T_0$. \end{theorem} \begin{theorem} Assume the hypotheses of Theorems \ref{t1} and \ref{t2}, and assume in addition that $h \perp \ker(-\Delta-\lambda_k)$, for some $k\ge 2$. Then there exists $\delta > 0$ such that for $\lambda \in (\lambda_k-\delta,\lambda_k)$ problem (\ref{p1}) has at least three solutions for $f = h + te_1$, with $t > T_0$. \end{theorem} The first solution of equation (\ref{p1}) is a negative solution, and its existence, for $t$ sufficiently large, is not difficult to prove (see \cite{D}, \cite{DJ}): first note that a negative solution satisfies the {\it linear equation} \begin{equation}\label{p2} \begin{gathered} -\Delta y - \lambda y = h + t e_1 \quad \mbox{in } \Omega\\ y = 0 \quad \mbox {on } \partial \Omega \end{gathered} \end{equation} The solution of this equation is unique, and we denote it, in dependence of $te_1$, by $\phi_t$ (we remark that if $\lambda = \lambda_k$ then $h \perp ker(-\Delta - \lambda_k)$ is required, and the solution is unique in $(ker (-\Delta - \lambda_k))^\perp$). Note that we may assume that $\int_\Omega h e_1 dx = 0$, and then the solution $\phi_t$ can be written as $\phi_t = w + s_t e_1$, with $\int_\Omega w e_1 dx = 0$ and $s_t = t/(\lambda_1 -\lambda)$ (and with $w \perp ker (-\Delta - \lambda_k)$ if $\lambda = \lambda_k$). Since $h \in L^r, r > N$, we have $w \in C^{1,1-N/r}({\overline \Omega})$, and it is known (see \cite{GT}) that on $\partial \Omega$ the (interior) normal derivative $\frac{\partial}{\partial n} e_1(x)$ is positive; hence $\phi_t < 0$ for $t$ sufficiently large. To find a second solution of equation (\ref{p1}), we set $u=v+\phi_t$; then $v$ solves \begin{equation}\label{p3} \begin{gathered} -\Delta v = \lambda v +(v+\phi_t )_+^{2^\star-1} +g(x,(v+\phi_t)_+) \quad \mbox {in } \Omega \\ v = 0 \quad \mbox{on } \partial \Omega \end{gathered} \end{equation} Clearly $v = 0$ is a solution of this equation, corresponding to the negative solution $\phi_t$ for equation (\ref{p1}). To find a second solution of equation (\ref{p3}) one can look for non trivial critical points of the functional \[ J(v) = \frac{1}{2}\int_\Omega ( |\nabla v|^2-\lambda v^2)dx -\frac{1}{2^\star}\int_\Omega (v+\phi_t)_+^{2^\star}dx -\int_\Omega G(x, (v+\phi_t)_+ )dx, \] where $G(x,s) : = \int_0^s g(x,\xi )\,d\xi$. This was the approach of de Figueiredo-Jianfu in \cite{DJ}. For applying the Generalized Mountain Pass theorem of Rabinowitz \cite{Ra} one needs to prove some geometric estimates. Furthermore, since the nonlinearities have critical growth, one needs to show that the minimax level avoids the non-compactness levels given by the ``concentrating sequences" $u_\epsilon$ (see \cite{BN} and below). In these estimates, the terms of the form $v + \phi_t + u_\epsilon$ are not easy to handle. In this paper we apply a method introduced in \cite{GR} to make such estimates easier. The idea consists in separating the supports of $v + \phi_t$ and $u_\epsilon$ by concentrating the support of the functions $u_\epsilon$ in small balls, and ``cutting small holes" into the functions $v + \phi_t$ such that the respective supports are disjoint. These manipulations create some errors, but these are easier to handle than to estimate the ``mixed terms" arising in expressions like $(v + \phi_t + u_\epsilon)_+^p$. Moving these ``small holes'' near $\partial \Omega$ where the first solution $\phi_t$ is small allows to further improve the estimates. \section{Variational setting and preliminary properties} We begin by replacing equation (\ref{p3}) by an approximate equation. We denote by $B_r(x_0) \subset \Omega$ a ball of radius $r$ and center $x_0 \in \Omega$. Choose $m \in \mathbb{N}$ so large that $B_{2/m}(x_0) \subseteq \Omega$, and let $\eta_m \in C_0^\infty(\Omega)$ such that $0 \le \eta_m(x) \le 1$, $|\nabla \eta_m (x)| \le 2m$ and \[ \eta_m(x)= \begin{cases} 0 & \mbox {in } B_{1/m}(x_0)\\ 1 & \mbox {in } \Omega \setminus B_{2/m}(x_0) \end{cases} \] Define the functions $\phi_t^m=\eta_m\phi_t$ and set \[ f_m = -\Delta\phi_t^m-\lambda\phi_t^m. \] Setting as before $u=v+\phi_t^m$ in equation (\ref{p1}), we see that then $v = \phi_t-\phi_t^m$ solves the equation \begin{equation}\label{p4} \begin{gathered} -\Delta v = \lambda v +(v+\phi_t^m)_+^{2^\star-1} +g(x,(v+\phi_t^m)_+) +( f-f_m ) \mbox{ in } \Omega \\ v= 0 \quad \mbox{on } \partial \Omega \end{gathered} \end{equation} Clearly $v=\phi_t-\phi_t^m$ corresponds to the trivial solution of this equation; for finding other solutions of (\ref{p4}) we look for critical points of the functional, $J:H \to \mathbb{R}$, \begin{align*} J(v)=&\frac{1}{2}\int_\Omega ( |\nabla v|^2-\lambda v^2) -\frac{1}{2^\star}\int_\Omega (v+\phi_t^m)_+^{2^\star}\\ &-\int_\Omega G(x, (v+\phi_t^m)_+ ) -\int_\Omega ( f-f_m )v \,, \end{align*} where $H$ denotes the Sobolev space $H=H_0^1(\Omega)$, equipped with the Dirichlet norm $\|u\|=(\int_\Omega |\nabla u|^2\, dx)^{1/2}$. We begin by estimating the ``error" given by the term $f - f_m$. \begin{lemma}\label{l1} For $N \ge 3$, as $m\to +\infty$ we have: \begin{gather}\label{fm0} \| \phi_t-\phi_t^m \| \le cm^{-N/2};\\ \label{fm} \Big| \int_\Omega ( f-f_m )\psi\,dx \Big| \le c \| \psi \| m^{-N/2}, \quad \mbox{for all } \psi \in H. \end{gather} \end{lemma} \paragraph{Proof.} Note first that by the regularity assumption $h \in L^r, r > N$, it follows that $\phi_t \in C^{1,1-N/r}({\overline \Omega})$, and hence in particular that there exists $c>0$ such that for any point $\bar{x} \in \partial \Omega$ \[ | \phi_t(x)| \le c|x-\bar{x}| . \] Furthermore we may choose for every large $m \in \mathbb{N}$ a point $x_0$ at distance $4/m$ from the boundary point $\bar{x}$, such that \begin{equation}\label{ptilde} | \phi_t(x)| \le \frac{ c_1}{m}, \quad \forall x \in B_{4/m}(x_0). \end{equation} We may assume that $x_0=0$ for every choice of $m$; from now on we write $B_r=B_r(0)$. Thus, we can estimate \begin{align*} \int_\Omega | & \nabla ( \phi_t-\phi_t^m ) |^2\\ =& \int_\Omega | \nabla \phi_t (1-\eta_m) -\phi_t \nabla \eta_m|^2\\ =&\int_{B_{\frac{2}{m}}} |\nabla \phi_t|^2|1-\eta_m|^2 -2\int_{B_{\frac{2}{m}}\setminus B_{\frac{1}{m}}} |\nabla \phi_t| ( 1-\eta_m) |\phi_t|\;|\nabla \eta_m| \\ &+\int_{B_{\frac{2}{m}}\setminus B_{\frac{1}{m}}} |\phi_t|^2|\nabla \eta_m|^2 \\ \leq& c_1 m^{-N}+c_2 m^{-N}+c_3 m^{-N} = c m^{-N} , \end{align*} hence (\ref{fm0}). The estimate (\ref{fm}) is now obtained as follows: \begin{align*} \Big| \int_\Omega ( f-f_m )\psi\,dx \Big| =& \Big| \int_\Omega |\nabla (\phi_t - \phi_t^m)\nabla \psi - \lambda (\phi_t - \phi_t^m)\psi dx \Big| \\ \leq& c \|\phi_t -\phi_t^m\| \; \| \psi \| \le c \| \psi \| m^{-N/2} \,, \end{align*} for all $\psi \in H$. \hfill %$\square$ Let $\lambda_1<\lambda_{2}\le \dots$ the eigenvalues of $-\Delta$ and $e_1,\,e_2\dots$, the corresponding eigenfunctions. Take $m$ as before and let $\zeta_m:\;\Omega\rightarrow\mathbb{R}$ be smooth functions such that $0 \le \zeta_m\le 1, |\nabla\zeta_m|\le4m$ and \[ \zeta_m(x)= \begin{cases} 0& \mbox{if } x\in B_{2/m}\\ 1& \mbox{if } x\in \Omega \setminus B_{3/m} \end{cases} \] We define ``approximate eigenfunctions" by $e_i^m=\zeta_me_i$. Then the following estimates hold. \begin{lemma}\label{l1a} As $m\to\infty$, we have $e_i^m\to e_i$ in $H$. Moreover, in the space $H^-_{j,m}=\mathop{\rm span}\{e_1^m,\dots,e_j^m\}$, we have \[ \max \Big\{\|u\|^2: u\in H^-_{j,m},\;\int_\Omega u^2=1\Big\} \le\lambda_j+c_jm^{-N} \] and \[ \int_\Omega \nabla e_i^m \nabla e_j^m dx = \delta_{ij} + O(m^{-N}). \] \end{lemma} \paragraph{Proof.} See \cite{GR} and observe that, since $\partial \Omega$ is of class $C^1$, also for the eigenfunctions $e_i$ an estimate as (\ref{ptilde}) holds. \hfill %$\square$ \smallskip Consider the family of functions \[ u_\varepsilon^\star(x)= \Big[ \frac{\sqrt{N(N-2)}\varepsilon}{\varepsilon^2+|x|^2}\Big]^{(N-2)/2} \] which are solutions to the equation \begin{gather*} -\Delta u = |u|^{2^\star-2} u \quad \mbox{in } \mathbb{R}^N \\ u(x) \to 0 \quad \mbox{as } |x| \to \infty \end{gather*} and which realize the best Sobolev embedding constant $H^1( \mathbb{R}^N) \subset L^{2^*}(\mathbb{R}^N)$, i.e. the value $$ S=S_N = \inf_{u \not= 0} \frac{\| u \|_{H}}{\|u \|_{L^{2^*}}} \, . $$ Let $\xi\in C^1_0(B_{1/m})$ be a cut--off function such that $\xi(x)=1$ on $B_{{1}/{2m}}$, $0\le\xi(x)\le 1$ in $B_{1/m}$ and $\| \nabla\xi \|_\infty\le 4m$. Let $u_\varepsilon(x):=\xi(x)u_\varepsilon^\star(x) \in H$. For $\varepsilon\to0$ we have the following estimates due to Brezis and Nirenberg \begin{lemma}[Brezis-Nirenberg, \cite{BN}] \label{brn} For fixed $m$ we have \begin{description} \item[(a)] $\|u_\varepsilon\|^2=S^{N/2}+O(\varepsilon^{N-2})$ \item[(b)] $\|u_\varepsilon\|^{2^\star}_{2^\star}=S^{N/2}+O(\varepsilon^{N})$ \item[(c)] $\|u_\varepsilon\|^2_2 \ge K_1\varepsilon^2+O(\varepsilon^{N-2}) $ \item[(d)] $\|u_\varepsilon \|_s^s\ge K_2\varepsilon^{N-\frac{N-2}{2}s}$. \end{description} Moreover, for $m \to \infty$ and $\varepsilon=o(1/m)$, we have (see \cite{GR}) \begin{description} \item[(e)] $\|u_\varepsilon\|^2=S^{N/2}+O((\varepsilon m)^{N-2})$ \item[(f)] $\|u_\varepsilon\|^{2^\star}_{2^\star}=S^{N/2}+O((\varepsilon m)^{N})$ \end{description} while (c) and (d) hold independently of $m$. \end{lemma} \paragraph{Proof.} We only prove (d); for the other estimates, see \cite{BN,GR,St}. \begin{align*} \int_{\Omega}u_\varepsilon^s\,dx \ge& c\int_{B_{\frac{1}{2m}}} \big( \frac{\varepsilon}{\varepsilon^2+|x|^2} \big)^{\frac{N-2}{2}s}\,dx \\ \ge & c\int_0^\varepsilon \big( \frac{\varepsilon}{\varepsilon^2+\rho^2} \big)^{\frac{N-2}{2}s} \rho^{N-1}d\rho \ge c\varepsilon ^{-\frac{N-2}{2}s}\varepsilon^N. \end{align*} which completes the proof. \hfill %$\square$ \section{The linking structure} In this section we prove that the functional $J$ has a ``linking structure" as required by the {\it Generalized Mountain Pass Theorem} by P. Rabinowitz \cite{Ra}. For the rest of this article, we assume $\lambda\in[\lambda_k,\lambda_{k+1})$. Let $H^+ = [\mathop{\rm span} \{ e_1, \dots , e_k \}]^\perp $, $ S_r = \partial B_r \cap H^+ $, $H^-_{m}=span\{e_1^m,\dots, e_k^m\}$ and $Q_m^\varepsilon = (B_R \cap H_m^-) \oplus [0,R]\{u_\varepsilon\}$, where $m \in \mathbb{N}$ is fixed. Define the family of maps ${\cal H} = \{ h : Q_m^\varepsilon \to H \ \hbox{continuous} : h \big|_{\partial Q_m^\varepsilon } = id \}$, and set \begin{equation}\label{c} \bar{c} = \inf_{h \in {\cal H}} \sup_{u \in h(Q_m^\varepsilon )} J(u) \end{equation} Then the Generalized Mountain Pass theorem of P. Rabinowitz states that if \begin{quote} 1) $J : H \to \mathbb{R}$ satisfies the Palais-Smale condition (PS) \\ 2) there exist numbers $0 < r < R$ and $\alpha_1 > \alpha_0$ such that \begin{gather}\label{1} J(v) \ge \alpha_1, \quad\mbox{for all } v \in S_r \\ \label{2} J(v) \le \alpha_0, \quad\mbox{for all } v \in \partial Q_m^\varepsilon\,, \end{gather} then the value $\bar{c}$ defined by (\ref{c}) satisfies $\bar{c} \ge \alpha_1$, and it is a critical value for $J$. \end{quote} First note that for $v\in H^-_m\oplus\mathbb{R}\{u_\varepsilon\}$, $v =w+s u_\varepsilon$, we have by definition $\mathop{\rm supp}(u_\varepsilon)\cap\mathop{\rm supp}(w)=\emptyset$. It is easy to prove that this implies that $$ J(v)\equiv J(w+s u_\varepsilon)=J(w)+J(s u_\varepsilon) \, . $$ We begin by showing that the functional $J$ satisfies condition (\ref{1}). \begin{lemma}\label{rr0} There exist numbers $r > 0$ and $\alpha_1>0$ such that \[ J(v)\ge\alpha_1\quad \mbox{for all } v \in S_{r }=\partial B_{r}\cap H^+ \] \end{lemma} \paragraph{Proof.} Let $v\in H^+$. From the variational characterization of $\lambda_{k+1}$ and the Sobolev embedding theorem, we have, using ($g_3$) and Lemma \ref{l1}, \begin{eqnarray*} J(v)&\ge & \frac{1}{2} \big(1-\frac{\lambda}{\lambda_{k+1}} \big) \int_\Omega |\nabla v|^2 - \frac{1}{2^\star} \int_\Omega v_+^{2^\star} - c \int_\Omega v_+^{p+1} -\int_\Omega ( f-f_m) v \\ &\ge & c_1\|v\|^2-c_2 \|v \|^{2^\star}-c_3 \|v \|^{p+1}- c_4 m^{-N/2} \|v\| \end{eqnarray*} Let $k_m(s)=c_1 |s|^2- c_2 |s|^{2^\star}-c_3|s|^{p+1}- c_4 m^{-N/2} |s|$. Clearly, there exists $m_0$ such that $\max_{\mathbb{R}}k_m(s)=M_m \ge M_{m_0} > 0$ for all $m \ge m_0$. Thus there exist $\alpha_1 > 0$ and $r > 0$ such that \[ J(v)\ge \alpha_1 > 0 \quad \mbox{for } \|v\|=r. \] which completes the proof. \hfill %$\square$ Next we prove condition (\ref{2}). \begin{lemma}\label{rr0bis} There exist $R > r$ and $\alpha_0 < \alpha_1$ such that for $\varepsilon$ sufficiently small \[ J\big|_{\partial Q_m^\varepsilon } < \alpha_0 \,. \] \end{lemma} \paragraph{Proof.} Let $v=w+s u_\varepsilon \in (H_m^- \cap {\bar B}_R) \oplus [0,R]\{u_\varepsilon]$. Since $J(v)=J(w)+J(s u_\varepsilon)$ we can estimate $J(w)$ and $J(s u_\varepsilon)$ separately. \[ J(w) \le \frac{1}{2} \int_\Omega | \nabla w |^2\,dx -\frac{\lambda}{2} \int_\Omega |w|^2\,dx -\frac{1}{2^\star} \int_\Omega ( w+\phi_t^m)_+^{2^\star}\,dx\,, \] since $\int_\Omega (f - f_m)w\,dx = \int_\Omega \nabla (\phi_t - \phi_t^m)\nabla w \,dx- \lambda \int_\Omega (\phi_t - \phi_t^m)w\,dx =0$. \[ \begin{array}{lcl} %\begin{multline*} \displaystyle J(s u_\varepsilon) &\le &\displaystyle\frac{s^2}{2} \int _{B_{\frac{1}{m}}}|\nabla u_\varepsilon|^2\,dx -\frac{\lambda s^2}{2}\int _{B_{\frac{1}{m}}} |u_\varepsilon|^2\,dx \\ &-&\displaystyle\frac{s^{2^\star}}{2^\star}\int _{B_{\frac{1}{m}}} |u_\varepsilon |^{2^\star}\,dx - \int _{B_{\frac{1}{m}}}f \; s u_\varepsilon \,dx. \end{array} \] %\end{multline*} Let $\partial Q_m^\varepsilon=\Gamma_1 \cup \Gamma_2 \cup \Gamma_3$, where %\begin{gather*} \[ \begin{array}{lcl} \Gamma_1 &=& \{v\in H : v=w+su_\varepsilon ,\; w\in H^-_m , \; \|w\|=R,\; 0\le s \le R \},\\ \Gamma_2 &=& \{v\in H : v=w+R u_\varepsilon,\; w\in H^-_m \cap \bar{B}_R \},\\ \Gamma_3 &=& H^-_m \cap \bar{B}_R\,. \end{array} \] %\end{gather*} Note that it follows by Lemma \ref{l1a} that \[ \int_\Omega | \nabla w |^2 \,dx\le ( \lambda_k + c_k m^{-N}) \int_\Omega |w|^2 \,dx, \quad \hbox{for all } w \in H^-_m \] 1. Suppose $v \in \Gamma_1$; then $ v=w+s u_\varepsilon$ with $\| w \| = R \;$ and $\;\, 0 \le s \le R$. \\ (i) if $\lambda \in (\lambda_k,\lambda_{k+1})$, we choose $m_0$ such that $c_k m^{-n}<\frac{\lambda-\lambda_k}{2}$, for $m\ge m_0$. Then, using Lemma \ref{brn} \begin{align*} J(v) \le & \frac{1}{2} \big( 1-\frac{\lambda}{\lambda_k+c_k m^{-N}}\big) \int_\Omega | \nabla w |^2 \,dx -\frac{1}{2^\star} \int_\Omega ( w+\phi_t^m)_+^{2^\star}\,dx\\ & + \frac{s^2}{2}\int_{B_{\frac{1}{m}}}|\nabla u_\varepsilon|^2\,dx -\frac{\lambda s^2}{2}\int_{B_{\frac{1}{m}}} u_\varepsilon^2\,dx -\frac{s^{2^\star}}{2^\star}\int_{B_{\frac{1}{m}}} u_\varepsilon^{2^\star}\,dx +s \| f \|_2 \| u_\varepsilon \|_2 \\ \le& -c R^2 + S^{N/2}\big[\frac{s^2}{2} -\frac{s^{2^\star}}{2^\star}\big] + R^{2^\star} O(\varepsilon^{N})+R^2 O(\varepsilon^{N-2} )+c s \\ \le&-c R^2 +c_1 +c_2 R +c_3 R^{2^\star}\varepsilon^{N-2}\,. \end{align*} Thus $J(v)\le 0$ for $R\ge R_0$ and $\varepsilon >0$ sufficiently small. \\ (ii) if $\lambda=\lambda_k$, for $w=R \bar{w} \in H_m^-$ with $\|\bar{w}\|=1$ we write $\bar{w}=\alpha y+\beta e_k^m$, with $y\in \mathop{\rm span}\{e_1^m,\dots,e_{k-1}^m\}$ and $\|y\|=1$. Then \[ J(w)=\frac{R^2}{2} \int_\Omega ( |\alpha \nabla y+\beta \nabla e_k^m|^2- \lambda_k|\alpha y+\beta e_k^m|^2) \,dx- \frac{R^{2^\star}}{2^\star} \int_\Omega \big(\bar{w}+\frac{\phi_t}{R} \big)_+^{2^\star} \,dx. \] Using Lemma \ref{l1a}, we can estimate the first integral as follows \begin{eqnarray*} \lefteqn{\int_\Omega ( |\alpha \nabla y+\beta \nabla e_k^m|^2- \lambda_k|\alpha y+\beta e_k^m|^2) \,dx}\\ &\le& \alpha^2\big(1-\frac{\lambda_k}{\lambda_{k-1}+c m^{-N}} \big) +\beta^2\big(1-\frac{\lambda_k}{\lambda_{k}+c m^{-N}} \big) \\ &&+ 2\alpha \beta \int_\Omega ( \nabla y \nabla e_k^m - \lambda_k y e_k^m )\,dx\\ &\le& -c \alpha^2 + c_1 ( \beta^2+2 \alpha \beta ) m^{-N}. \end{eqnarray*} Note now that if $|\alpha|\ge \delta>0$, for some $\delta>0$, then \[ J(w)\le -\frac{c \delta^2 R^2}{2} + c_1 m^{-N} \] and hence $J(v)\le 0$ for $R\ge R_1(\delta)$. We show now that there exists $\delta > 0$ such that if $|\alpha|\le \delta$, then there exist constants $c_2>0$ and $R_2 > 0$ such that \[ \int_{\Omega} \big( \bar{w}+\frac{\phi_t}{R}\big)_+^{2^\star}\ge c_2>0 \] for all $R \ge R_2$ and for all $w\in H^-_m\cap \partial B_R$. To this aim we prove that there exist $\delta > 0$ and $\eta > 0$ such that \[ \max_{\bar{\Omega}} ( \alpha y+\beta e_k^m ) \ge \eta > 0 \quad \hbox{for all } y\in H^-_{k-1,m},\,\|y\|=1,\; |\alpha|\le \delta \,. \] By contradiction assume that there exist sequences $|\alpha_n| \le 1/n$, $y_n \in H_{k-1,m}^-$ with $\|y_n\|=1$ such that \[ \max_{\bar \Omega} \{\alpha_n y_n+\beta_n e_k^m \} \to 0 \quad \mbox{as} \quad n\to+\infty. \] Then $\alpha_n y_n\to 0$, $\beta^2_n=1-\alpha_n^2 + O(m^{-N})\to \beta^2 = 1 + O(m^{-N}) \ge 1/2$, for $m \ge m_0$. Therefore, we conclude that \[ \max_{\bar \Omega} ( \beta e_k^m) = 0 \ , \mbox{ with } \beta^2 \ge \frac 12 \mbox{ for } m \ge m_0 \mbox{ i.e. }\ ( e_k^m)^+=0 \,. \] This is a contradiction since $e_k^m\to e_k$ in $H$ implies that $e_k^m$ must change sign, for $m$ large. Therefore, there exist $ \delta > 0$, $\eta > 0$ such that \[ \max_{\bar \Omega} \left\{ {\bar w}, |\alpha| \le \delta \right\} \ge \eta > 0 \ , \ \forall {\bar w} \in H_{k,m}^- , \|{\bar w}\| = 1, m \ge m_0 \,. \] Denoting $\Omega_{\bar{w}}=\{ x\in \Omega:\; \bar{w}(x)\ge\ \eta/2 \}$, then $|\Omega_{\bar w}| \ge \nu > 0$, $\forall {\bar w} \in H_{k,m}^-$, with $ \|{\bar w}\| = 1$ and $|\alpha|\le \delta$, $m \ge m_0$, since the functions ${\bar w} \in H_{k,m}^-$ are equicontinuous. Moreover \[ \frac{\phi_t}{R}>-\frac{\eta}{4} \quad \mbox{for }R \mbox{ sufficiently large}\,. \] Then \[ \int_{\Omega} ( \bar{w}+\frac{\phi_t}{R})_+^{2^\star}\ge ( \frac{\eta}{4} )^{2^\star} | \Omega_{\bar{w}} |. \] Thus, we can conclude that there exists $R_2 > 0$ such that \[ J(w) \le c R^2 - R^{2^\star} \int_\Omega ({\bar w} + \frac{\phi_t }{R})_+^{2^*} \le c R^2 - R^{2^\star} (\frac{\eta}{4})^{2^*} \; \nu \le 0 \ , \] for all $R\ge R_2$. In particular $J(w)\to -\infty$ as $R\to +\infty$. \smallskip \noindent 2. Let $v\in\Gamma_2$, i.e. $v = w + R u_\varepsilon$ with $ \| w \| \le R$. Then \begin{align*} J(v) =& J(w)+J(R u_\varepsilon)\\ \le& c m^{-N} \| w \|^2 +\frac{R^2}{2}\int_{B_{\frac{1}{m}}}|\nabla u_\varepsilon|^2\,dx -\frac{R^{2^\star}}{2^\star}\int_{\Omega} u_\varepsilon^{2^\star}\,dx + R \| f \|_2 \| u_\varepsilon \|_2 < 0 \,, \end{align*} for $R$ sufficiently large. Now fix $R>0$ such that the previous estimates hold. \smallskip \noindent 3. Let $v \in \Gamma_3$, i.e. $ v = w \in H^-_m\cap B_R$. Hence \[ J(v) \le c_1 m^{-N} \|w\|^2-\frac{1}{2^\star}\int_\Omega( w+\phi_t )_+^{2^\star} \le \alpha_0 \] if $m$ is sufficiently large. \hfill %$\square$ \section{Existence of a second solution} In this section we prove Theorems \ref{t1} and \ref{t2}. By the Linking Theorem we construct a Palais--Smale sequence $\{v_n\}\subset H$ at the minimax level $\bar{c}$; the sequence $\{v_n\}$ satisfies \begin{equation} \begin{aligned} J(v_n) = &\frac{1}{2}\int_\Omega( |\nabla v_n|^2-\lambda v_n^2 )\,dx -\frac{1}{2^\star}\int_\Omega( v_n+\phi_t^m )_+^{2^\star}\,dx \\ &-\int_\Omega G(x, (v_n+\phi_t^m )_+)\,dx +\int_\Omega (f-f_m)v_n\,dx =\bar{c}+o(1) \end{aligned} \label{p_1} \end{equation} and \begin{equation} \begin{aligned} \langle J^\prime(v_n),z\rangle = & \int_\Omega( \nabla v_n \nabla z-\lambda v_n z)\,dx -\int_\Omega( v_n+\phi_t^m )_+^{2^\star-1} z\,dx \\ &-\int_\Omega g(x,(v_n+\phi_t^m )_+)z\,dx +\int_\Omega (f-f_m)z\,dx =o(1)\|z\| \end{aligned} \label{p_2} \end{equation} for all $z\in H$. \newpage \begin{lemma}\label{nn0} Under the hypotheses of Theorem \ref{t1} or Theorem \ref{t2}, the sequence $\{v_n\}$ is bounded in $H$. \end{lemma} \paragraph{Proof.} >From (\ref{p_1}) and (\ref{p_2}), it follows that \begin{eqnarray*} \lefteqn{ J(v_n)-\frac{1}{2}\langle J^\prime(v_n),v_n\rangle }\\ &=& \frac{1}{N}\int_\Omega( v_n+\phi_t^m )_+^{2^\star} -\frac{1}{2} \int_\Omega( v_n+\phi_t^m)_+^{2^\star-1} \phi_t^m\\ && - \int_\Omega G(x, (v_n+\phi_t^m)_+) +\frac{1}{2}\int_\Omega g(x, (v_n+\phi_t^m )_+)v_n - \frac{1}{2}\int_\Omega( f-f_m)v_n\\ &=& \bar{c}+o(1)+o(1)\|v_n\| \end{eqnarray*} Therefore, using that $\phi_t^m\le 0$ and Lemma 4, \begin{eqnarray*} \frac{1}{N} \int_\Omega (v_n+\phi^m_t )_+^{2^\star} \,dx & \le & \int_\Omega G(x, (v_n+\phi_t^m )_+)\,dx \\ &&-\frac{1}{2} \int_\Omega g(x, (v_n+\phi^m_t )_+)(v_n+\phi^m_t)\,dx\\ &&+c +( o(1)+dm^{-N} ) \|v_n\| \end{eqnarray*} Then by ($g_3$), we get \begin{eqnarray*} \lefteqn{\int_\Omega (v_n+\phi^m_t )_+^{2^\star} \,dx}\\ & \le & c_1 \int_\Omega (v_n+\phi^m_t )_+^{p+1} \,dx +{c}+( o(1)+dm^{-N} ) \|v_n\|\\ & \le & c_1 \Big( \int_\Omega (v_n+\phi^m_t )_+^{2^\star} \,dx \Big)^{\frac{p+1}{2^\star}}+c+( o(1)+dm^{-N} ) \|v_n\| \end{eqnarray*} Since $p+1<2^\star$, we obtain \begin{equation} \int_\Omega (v_n+\phi_t )_+^{2^\star}\,dx \le {c}+(o(1)+dm^{-N} ) \|v_n\| \le c_1+c_2 \|v_n\| \label{pp4} \end{equation} $(i)$ First we consider the case $\lambda\in(\lambda_k,\lambda_{k+1})$. Let $v_n=v_n^+ + v_n^-$ (as in \cite{DJ}), with $v_n^- \in H^-_k=\mathop{\rm span}\{ e_1 \dots e_k\}$ and $v_n^+\in (H^-_k)^{\perp}$. We obtain \begin{eqnarray*} \langle J^{\prime}(v_n),v_n^+\rangle & = & \int_\Omega ( | \nabla v_n^+|^2 -\lambda ( v_n^+)^2 )\,dx -\int_\Omega ( v_n + \phi_t^m )_+^{2^\star -1} v_n^+\,dx \\ & &- \int_\Omega g( x,( v_n + \phi_t^m )_+ ) v_n^+\,dx -\int_\Omega( f-f_m) v_m^+ dx =o(1) \| v_n^+ \| \end{eqnarray*} \newpage \noindent From the variational characterization of $\lambda_{k+1}$ we get, using the H\"{o}lder and Young inequalities and (\ref{pp4}), \begin{eqnarray*} \lefteqn{ \big( 1-\frac{\lambda}{\lambda_{k+1}} \big) \| v_n^+ \|^2}\\ & \le & \int_\Omega ( v_n + \phi_t^m )_+^{2^\star -1} v_n^+ +c\int_\Omega( v_n +\phi_t^m )_+^p | v_n^+| + o(1) \| v_n^+ \| +dm^{-N} \| v_n^+ \|\\ &\le & \varepsilon \Big( \int_\Omega| v_n^+ |^{2^\star} \Big)^{2/2^\star}+ c_\varepsilon( \int_\Omega( v_n +\phi_t^m )_+^{2^\star} )^{\frac{2(2^\star-1)}{2^\star}} \\ && +c\Big(\int_\Omega( v_n +\phi_t^m )_+^{2^\star} \Big)^{p/2^\star} \Big(\int_\Omega | v_n^+ |^{\frac{2^\star}{2^\star-p}} \Big)^{\frac{2^\star-p}{2^\star}}+ c\| v_n^+ \| \\ & \le& \varepsilon \| v_n^+ \|^2 + c_\varepsilon\Big( \int_\Omega( v_n +\phi_t^m )_+^{2^\star} \Big)^{\frac{2(2^\star-1)}{2^\star}}+ \varepsilon\Big(\int_\Omega | v_n^+ |^{\frac{2^\star}{2^\star-p}} \Big)^{2\frac{2^\star-p}{2^\star}} \\ && +c_\varepsilon\big( \int_\Omega\ ( v_n +\phi_t^m )_+^{2^\star} \Big)^{2 p/2^\star}+c\| v_n^+ \| \end{eqnarray*} By (\ref{pp4}) and by the Sobolev embedding theorems, we obtain \begin{equation} \| v_n^+ \|^2 \le {c} + o(1) \big(\| v_n \|^{\frac{N+2}{N}} +\| v_n \|^{\frac{2 p}{2^\star}} \big) +c \| v_n^+ | | \label{eqmeno} \end{equation} For $v_n^-\in H^-$, we have \begin{eqnarray*} \lefteqn{ ( \frac{\lambda}{\lambda_k}-1 ) \int_\Omega | \nabla v^-_n |^2}\\ &\le& \int_\Omega\ ( v_n +\phi_t^m )_+^{2^\star-1} | v^-_n | + \int_\Omega\ g(x,( v_n +\phi_t^m )_+) | v^-_n | + \int_\Omega |f-f_m| |v_n^-| \end{eqnarray*} In the same way we obtain \begin{equation} \| v_n^- \|^2_{H^1} \le \bar{c} + o(1) \big( \| v_n \|^{\frac{N+2}{N}} + \| v_n \|^{\frac{2 p}{2^\star}} \big) +c \| v_n^- \| \label{eqpiu} \end{equation} Joining (\ref{eqmeno}) and (\ref{eqpiu}), we find \[ \|v_n\|^2\le c+c ( \|v_n\|^{\frac{N+2}{N}} + \|v_n\|^{\frac{2p}{2^\star}} )+c \|v_n\|, \] so $v_n$ is bounded in $H$.\smallskip \noindent $(ii)$ If $\lambda=\lambda_k$ we write $v_n=v_n^-+v_n^+ +\beta_n e_k=w_n+\beta_n e_k$, where we denote with $v_n^-$ and $v_n^+$ the projections of $v_n$ onto the subspace $H^-_{k-1}=\mathop{\rm span}\{e_1,\dots,e_{k-1} \}$ and $H^+_k=(H^-_k)^\perp$ respectively. With a similar argument as above we obtain the following estimate \begin{equation} \|w_n \|^2\le c+c \big( \| v_n \|^{\frac{N+2}{N}}+\| v_n \|^{\frac{2p}{2^\star}}\big) +c\| w_n \| \label{pp5} \end{equation} We can assume $\|v_n\|\ge 1$. Then, from (\ref{pp5}), we have \begin{equation} \| w_n \|^2\le c+c \| v_n \|^{\frac{2p}{2^\star}}\le c+c ( \| w_n \|+| \beta_n |)^{\frac{2p}{2^\star}} \label{pstar} \end{equation} If $\beta_n$ is bounded we conclude as above. If not, we may assume $\beta_n \to+\infty$ and $\|w_n\|\to+\infty$ (we neglect the case $\|w_n\|\le c$ which is much easier). Therefore, from (\ref{pstar}), \[ \| w_n \|^2\le \big[ \frac{1}{2} ( \| w_n \|+\beta_n^2)\big]^{p/2^\star}. \] Then $\| w_n \|\le c \beta_n^{p/2^\star}$ and $ \| \frac{w_n}{\beta_n}\|\to 0$ since $\frac{p}{2^\star}<1$. Therefore, possibly up to a subsequence, $w_n/\beta_n\to 0$ a.e. and strongly in $L^q$, $2\le q<2^\star$. Therefore, for all $q\in (2,2^\star)$ \begin{equation} \int_\Omega ( \frac{w_n+\phi_t^m}{\beta_n}+e_k )^q_+ e_k\,dx \quad\to\quad \int_\Omega( e_k )^{q+1}_+ \,dx. \label{pint} \end{equation} Moreover, since \begin{eqnarray*} o(1)& =&\langle J^\prime(v_n), e_k\rangle \\ &=& -\int_\Omega (v_n+\phi_t^m)^{2^\star-1}_+ e_k -\int_\Omega g(x,(v_n+\phi_t^m )_+) e_k +\int_\Omega (f-f_m) e_k \end{eqnarray*} we get, using $(g_3)$ \[ \int_\Omega \big( \frac{w_n+\phi_t^m}{\beta_n}+e_k\big) ^{2^\star-1}_+ e_k\le o(1)+\frac{c}{\beta_n^{2^\star-1-p}} \int_\Omega \big( \frac{w_n+\phi_t^m}{\beta_n}+e_k\big)_+^p e_k. \] Finally, by (\ref{pint}) we get \[ \int_\Omega (e_k )_+^{2^\star}\le 0 \] which is a contradiction. Thus $(v_n)$ is bounded. \hfill %$\square$ Returning to relation (\ref{p_2}), we may therefore assume, as $n\to +\infty$: $v_n \rightharpoonup v$ weakly in $H^1_0$, $v_n \to v$ in $L^q \quad 2\le q< 2^\star$ and $v_n \to v$ a. e. in $\Omega$. In particular, it follows that $v$ is a weak solution of \begin{equation} \begin{gathered} -\Delta v =\lambda v+( v +\phi_t^m )_+^{2^\star-1}+g(x,( v +\phi_t^m )_+) +f-f_m \quad\mbox{in } \Omega\\ v =0 \quad \mbox{on } \partial \Omega \end{gathered}\label{eqcz} \end{equation} To conclude the proof, It remains to show that $v \ne \phi_t-\phi_t^m$, the ``trivial'' solution of (\ref{eqcz}). First, we estimate \begin{eqnarray*} \lefteqn{ J(\phi_t-\phi_t^m) }\\ &= &\frac{1}{2}\int_\Omega | \nabla (\phi_t-\phi_t^m)|^2 -\frac{\lambda}{2}\int_\Omega| \phi_t-\phi_t^m|^2 -\int_\Omega(f-f_m)(\phi_t-\phi_t^m) \\ &= & -\frac{1}{2} \int_\Omega \big[ | \nabla (\phi_t-\phi_t^m)|^2 -\lambda |\phi_t-\phi_t^m|^2 \big]. \end{eqnarray*} \vspace{5mm} From Lemma \ref{l1} we get, taking $\varepsilon^\beta=\frac{1}{m}\quad 0<\beta<1$, \begin{equation} |J(\phi_t-\phi_t^m)|\le c m^{-N}:=c \varepsilon^{\beta N}. \label{eqplus} \end{equation} Since $v$ is a weak solution of (\ref{eqcz}), we have \begin{eqnarray*} \int_\Omega ( | \nabla v |^2 -\lambda v^2)\,dx - \int_\Omega ( v +\phi_t^m )_+^{2^\star}\,dx + \int_\Omega ( v +\phi_t^m )_+^{2^\star-1} \phi_t^m\,dx&&\\ -\int_\Omega g(x,( v +\phi_t^m )_+) v\,dx -\int_\Omega ( f-f_m ) v\,dx&=&0 \end{eqnarray*} By the Brezis--Lieb Lemma (\cite{BL}) \begin{equation} \int_\Omega ( v_n +\phi_t^m )_+^{2^\star}\,dx = \int_\Omega ( v_n -v )_+^{2^\star}\,dx + \int_\Omega ( v +\phi_t^m )_+^{2^\star}\,dx +o(1) \label{eq1st} \end{equation} Since $v_n \to v$ in $L^q \quad 2\le q < 2^\star$, we have \begin{equation} \int_\Omega G(x,( v_n +\phi_t^m )_+)\,dx - \int_\Omega G(x,( v +\phi_t^m )_+)\,dx =o(1). \label{eq2st} \end{equation} Since $v_n \rightharpoonup v$ in $H$, \begin{equation} \int | \nabla v_n | ^2 = \int | \nabla v |^2+\int| \nabla (v_n-v )|^2+o(1). \label{eq3st} \end{equation} Then, by (\ref{p_1}), (\ref{eq1st}) and (\ref{eq3st}), we find \begin{eqnarray} \bar{c}+o(1)& =&J( v_n ) \label{p2palle} \\ & =& J( v )+\frac{1}{2} \int_\Omega |\nabla( v_n -v )|^{2}\,dx - \frac{1}{2^\star} \int_\Omega ( v_n -v )_+^{2^\star}\,dx +o(1) \nonumber \end{eqnarray} Similarly, since $J^\prime(v)=0$, we obtain \begin{eqnarray*} \langle J^\prime( v_n ),v_n\rangle & =& \int_\Omega |\nabla( v_n -v)|^{2} - \int_\Omega ( v_n -v )_+^{2^\star} - \int_\Omega ( v_n -v)_+^{2^\star-1} \phi_t^m \\ &&- \int_\Omega g(x,( v_n +\phi_t^m)_+) v_n + \int_\Omega g(x,( v +\phi_t^m)_+) v +o(1). \end{eqnarray*} Since \[ \int_\Omega ( v_n -v )_+^{2^\star-1} \phi_t^m\,dx=o(1) \] and \[ \int_\Omega g(x,( v_n +\phi_t^m )_+) v_n\,dx - \int_\Omega g(x,( v +\phi_t^m )_+) v\,dx =o(1), \] we get \begin{equation} \int_\Omega |\nabla( v_n -v )|^{2}\,dx= \int_\Omega ( v_n -v )_+^{2^\star}\,dx +o(1). \label{ptilde2} \end{equation} Now, let \[ K=\lim_{n\to+\infty} \int_\Omega |\nabla( v_n -v )|^{2}\,dx. \] If $K=0$, then $v_n\to v$ strongly in $H$ and in $L^{2^\star}$; then by (\ref{p2palle}) and (\ref{eqplus}) \[ J(v)=\bar{c}\ge \alpha_1>c \varepsilon^{\beta N} \ge J ( \phi_t - \phi_t^m ) \] so that $v\ne \phi_t-\phi_t^m$. If $K>0$, using the Sobolev inequality and (\ref{ptilde2}), we have (as in \cite{DJ}) \begin{eqnarray*} \| v_n -v \|^{2} & \ge & S \Big( \int_\Omega | v_n-v|^{2^\star}\,dx \Big)^{2/2^\star} \ge S \Big( \int_\Omega ( v_n-v)_+^{2^\star}\,dx \Big)^{2/2^\star} \\ & \ge & S \Big( \int_\Omega |\nabla ( v_n-v)|^{2}\,dx+o(1) \Big)^{2/2^\star} \end{eqnarray*} This implies that $K\ge S K^{\frac{N-2}{N}}$, that is $K\ge S^{N/2}$. To complete the proof we use the following Lemmas which will be proved below. \begin{lemma}\label{ma1} Under the hypotheses of Theorem \ref{t1} one has, for $\varepsilon^\beta=1/m$, with $\alpha/N < \beta< (N-4)/(N-2)$, \begin{equation} \bar{c} < \frac {1}{N} S^{N/2} -c \varepsilon^2. \label{pp2} \end{equation} \end{lemma} \begin{lemma}\label{ma2} Suppose that the hypotheses of Theorem \ref{t2}2 are satisfied. Then for $\varepsilon^\beta=1/m$, with \[\max \big\{ 1-\frac{N-2}{2N}(q+1), \frac{2(N-1)}{N-2}-(q+1) \big\} <\beta<\frac{1}{2}(q+1)-\frac{2}{N-2}, \] we have \[ \bar{c} < \frac{1}{N} S^{N/2}-c \varepsilon^{N-\frac{N-2}{2}(q+1)}. \] \end{lemma} By (\ref{p2palle}) and (\ref{ptilde2}) we get \[ J(v)+\frac{K}{N} =\bar{c} \le \frac{1}{N}S^{N/2}- \begin{cases} c \varepsilon^2 & \mbox{(Theorem \ref{t1})}\\ c \varepsilon^{N-\frac{N-2}{2}(q+1)} &\mbox{(Theorem \ref{t2})} \end{cases} \] Assume now by contradiction that $v\equiv \phi_t-\phi_t^m$. Then we get by (\ref{eqplus}) \[ \frac{K}{N} + J(v) \ge \frac{1}{N}S^{N/2}-c \varepsilon^{\beta N} \] which is impossible, due to the choice of $\beta$. \hfill %$\square$ \paragraph{Proof of Lemma \ref{ma1}.} Let $\varepsilon>0$ and $m \in \mathbb{N}$ be fixed such that the hypotheses of the Linking Theorem are satisfied. For $v=w+s u_\varepsilon$, we have \begin{eqnarray*} J(v) &= &J(w) + J(s u_\varepsilon) \\ &\le& J(w) + \frac{s^2}{2}\int_{B_{\frac{1}{m}}}| \nabla u_\varepsilon |^2 - \frac{s^{2^\star}}{2^\star}\int_{B_{\frac{1}{m}}}u_\varepsilon^{2^\star} -\frac{\lambda s^2}{2}\int_{B_{\frac{1}{m}}}u_\varepsilon^{2}\\ &&+ s \int_{B_{\frac{1}{m}}} |f - f_m|\; |u_\varepsilon | \end{eqnarray*} As above, we have $J(w)\le c m^{-N}$. To estimate the last inequality we use the argument developed in \cite{GR} (Lemma \ref{l1}). We have from ({\it{e}}) and ({\it{f}}) in Lemma \ref{brn} \begin{eqnarray*} \frac{s^2}{2} \int_{B_{1/m}} | \nabla u_\varepsilon |^2 - \frac{s^{2^\star}}{2^\star} \int_{B_{1/m}} u_\varepsilon ^{2^\star} &\le&( \frac{s^2}{2}- \frac{s^{2^\star}}{2^\star} ) \big[ S^{N/2}+O( (\varepsilon m)^{N-2} ) \big] \\ &\le& ( \frac{1}{2}- \frac{1}{2^\star} ) \big[ S^{N/2}+O( (\varepsilon m)^{N-2} ) \big] \\ &\le&\frac{S^{N/2}}{N} + c (\varepsilon m)^{N-2}. \end{eqnarray*} (since $\frac{s^2}{2}- \frac{s^{2^\star}}{2^\star}$ attains its maximum at $s=1$) We make use of the H\"{o}lder inequality to estimate the last term. Let $\alpha$ be such that $\frac{1}{\alpha}+\frac{1}{r}+\frac{1}{2}=1$, i.e. $\frac{1}{\alpha} = \frac{1}{2}-\frac{1}{r} > \frac{1}{2}-\frac{1}{N}=\frac{1}{2^\star}$, in particular $2^\star>\alpha$. Since $\mathop{\rm supp} f_m \subseteq \Omega \backslash B_{1/m}$ we have \begin{eqnarray*} \int_{B_{1/m}} | f-f_m |\, | u_\varepsilon | &\le& \Big( \int_{B_{1/m}} | f |^r \Big)^{1/r} \Big( \int_{B_{1/m}} | u_\varepsilon |^2 \Big)^{1/2} ( \mu ( B_{1/m} ) )^{{1}/{\alpha}}\\ &\le& c \varepsilon m^{-N/\alpha}. \end{eqnarray*} If $\varepsilon^\beta=1/m$ with $0<\beta<1$ we have, by ({\it {c}}) in Lemma \ref{brn}, \begin{equation} \begin{aligned} J(v) \le & \frac{1}{N} S^{N/2} + c_1 m^{-N}+ c_2 (\varepsilon m)^{N-2}-c_3 \varepsilon^2+ c_4 \varepsilon m^{N/\alpha} \\ = & \frac{1}{N} S^{N/2} + c_1 \varepsilon^{N \beta} +c_2 \varepsilon^{(1- \beta)(N-2)} -c_3 \varepsilon^2 + \varepsilon^{1+ \beta N/\alpha} \end{aligned} \label{ri3} \end{equation} We choose $\beta$ such that \[ \frac{\alpha}{N} < \beta < 1 - \frac{2}{N-2}\,. \] Such a choice is possible only for $N \ge 6$. This then implies that for $\varepsilon > 0$ sufficiently small \[ J(v)<\frac{1}{N} S^{N/2} - c \varepsilon^2, \] in particular $\bar{c}<\frac{1}{N} S^{N/2} - c \varepsilon^2$. \hfill %$\square$ \paragraph{Proof of Lemma \ref{ma2}.} With a similar argument as above and using ($g_4$) and Lemma \ref{brn}(\textit{d}), we obtain \begin{eqnarray*} J(v)& = & J(w) +J(s u_\varepsilon)\\ &\le& J(w) + \frac{s^2}{2}\int_\Omega | \nabla u_\varepsilon|^2\,dx -\frac{\lambda s^2}{2}\int_\Omega u_\varepsilon^2\,dx -\frac{s^{2^\star}}{2^\star}\int_\Omega u_\varepsilon^{2^\star} \,dx\\ &&-\int_\Omega G(x,( s u_\varepsilon +\phi_t^m)_+ )\,dx -s \int_\Omega ( f-f_m) u_\varepsilon\,dx \end{eqnarray*} Reasoning as in Lemma \ref{ma1} from ({\it{d}}) in Lemma \ref{brn} we obtain as in (\ref{ri3}) \begin{eqnarray*} J(v)& \le & \frac{S^{N/2}}{N}+c_1 \varepsilon^{N\beta} +c_2 \varepsilon^{(1-\beta)(N-2)} - c_3 s^{q+1} \int_\Omega u_\varepsilon^{q+1}\,dx + c_4 \varepsilon^{1+\beta N/\alpha}\\ & \le& \frac{S^{N/2}}{N}+c_1 \varepsilon^{N\beta} +c_2 \varepsilon^{(1-\beta)(N-2)} -c_3 \varepsilon^{-\frac{N-2}{2}(q+1)+N} + c_4 \varepsilon^{1+\beta N/\alpha} \end{eqnarray*} Using that $\alpha<2^\star$, we get the following condition on $\beta\in(0,1)$ \[ N-\frac{N-2}{2} (q+1)< \begin{cases} \beta N \\ 1+\frac{\beta N}{2^\star}\\ (1-\beta) (N-2) \end{cases} \] which is equivalent to the system \begin{gather*} 1-\frac{N-2}{2N}(q+1) < \beta < - \frac{2}{N-2} + \frac{1}{2}(q+1)\\ \frac{2(N-1)}{N-2}- (q+1) < \beta < - \frac{2}{N-2} + \frac{1}{2}(q+1) \end{gather*} This choice is possible if \[ q+1 > \begin{cases} \frac{N^2}{(N-1)(N-2)}\\ \frac{2}{3} \cdot \frac{2N}{N-2} \end{cases} \] Therefore the result follows. \hfill %$\square$ \section{Existence of a third solution} We consider only the case $N\ge 6$ and $g\equiv 0$; the other cases follow with small changes. Let $\phi_t(\lambda)$ denote the negative solution of (1) for $\lambda\in (\lambda_k-\delta,\lambda_k)$. Since $h\in \ker (-\Delta-\lambda_k)^\perp$, $\phi_t(\lambda)$ is uniformly bounded when $\lambda\to \lambda_k$. Consider the functional \[ J_\lambda(v)= \frac{1}{2} \int_\Omega ( |\nabla v |^2-\lambda v^2 )\,dx -\frac{1}{2^\star} \int_\Omega ( v+\phi_t(\lambda))_+^{2^\star}\,dx -\int_\Omega ( f-f_m )v\,dx. \] We prove the main geometrical properties of the functional $J_\lambda$. \begin{lemma}\label{le7} Let $H^+=\mathop{\rm span}\{e_1,\dots,e_k\}^\perp$. There exist $\bar{\alpha}>0$, $\bar{r}>0$ such that \[ J_\lambda(v)\ge\bar{\alpha}>0\quad\mbox{for all } v\in S_{\bar{r}}=\partial B_{\bar{r}}\cap H^+, \; \lambda\in (\lambda_{k-1},\lambda_k ). \] \end{lemma} \paragraph{Proof.} With a similar argument as in Lemma \ref{rr0}, since $\lambda<\lambda_k$, we have \begin{eqnarray*} J_\lambda(v)&\ge& \frac{1}{2} \big( 1-\frac{\lambda}{\lambda_{k+1}} \big) \|v\|^2 -c_2 \|v\|^{2^\star}-c_3 m^{-N/2} \|v\| \\ & \ge& \frac{1}{2} \big( 1-\frac{\lambda_k}{\lambda_{k+1}} \big) \|v\|^2 -c_2 \|v\|^{2^\star}-c_3 m^{-N/2} \|v\| \end{eqnarray*} Then there exist $m_0$, $\bar{\alpha}$ and $\bar{r}$ such that for $m\ge m_0$, \[ J_\lambda(v)\ge \bar{\alpha}>0 \] for $\|v\|=\bar{r}$ and all $\lambda\in( \lambda_{k-1},\lambda_k )$. \hfill %$\square$ Now let $H_m^-=\mathop{\rm span} \{e_1^m,\dots,e^m_k\}$ as in the proof of Theorems \ref{t1} and \ref{t2}. \begin{lemma}\label{ma3} Let $Q_m=(H^-_m\cap B_R(0))\oplus[0,R] u_\varepsilon$, where $\varepsilon^\beta=1/m $. Then, if $R$ is large enough, there exists $\bar{m}\ge m_0$ such that:\\ (i) $J_{\partial Q_{\bar{m}}}<\bar{\alpha}$\\ (ii) $ \sup_{Q_{\bar{m}}} J<\frac{S^{N/2}}{N}-c \bar{\varepsilon}^2$. \end{lemma} \paragraph{Proof.} (i) Let $\partial Q_{\bar{m}}=\Gamma_1\cup\Gamma_2\cup\Gamma_3$ as in Lemma \ref{rr0bis}. \noindent (1) For $v=w+s u_\varepsilon\in\Gamma_1$ we have $J(v)=J(w)+J(s u_\varepsilon)$. Arguing as in Lemma \ref{rr0bis} (ii) we have \begin{equation} J(s u_\varepsilon) \le c_3 R^{2^\star} \varepsilon^{N-2}+c_1+c R \label{eqr1} \end{equation} Writing $w=R \bar{w}\in H_m^-$ with $\| \bar{w} \|=1$, $\bar{w}=\alpha y+\beta e^m_k$, $y\in \mathop{\rm span}\{e_1^m,\dots,e_{k-1}^m\}$, $\|y\|=1$, it was shown in Lemma \ref{rr0bis} (ii) that there exists a number $\sigma>0$ such that for $|\alpha|^2\le \sigma$, \begin{equation} J(w) \le c_1 R^2-c_2 R^{2^\star}. \label{eqr2} \end{equation} On the other hand, if $|\alpha|^2 > \sigma$, one deduces as in Lemma \ref{rr0bis} (ii) (using that $\delta < \lambda_k-\lambda_{k-1}$) that \begin{eqnarray*} J(w) & \le & \frac{R^2}{2} \int ( \alpha\nabla y+\beta\nabla e^m_k )^2 -\lambda | \alpha y+\beta e^m_k|^2 \\ & \le & \frac{R^2}{2} \Big[ \alpha^2 ( 1-\frac{\lambda}{\lambda_{k-1}+c m^{-N}} ) +\beta^2 ( 1-\frac{\lambda}{\lambda_{k}+c m^{-N}} ) \\ & & +2 \alpha \beta \int ( \nabla y \nabla e^m_k - \lambda y e^m_k ) \Big] \\ & \le & \frac{R^2}{2} \Big[ \sigma \frac{\lambda_{k-1}-( \lambda_k-\delta ) +c m^{-N}}{\lambda_{k-1}+c m^{-N}} +\beta^2 \frac{\delta+c m^{-N}}{\lambda_k+c m^{-N}} +2 \alpha \beta c m^{-N} \Big] \\ & \le & R^2 [-c \sigma+c_1 \delta+c_2 m^{-N}]. \end{eqnarray*} Thus, we get in this case, for $\delta$ sufficiently small and $m$ sufficiently large \begin{equation} J(w)\le -\frac{c}{2} \sigma R^2. \label{eqr3} \end{equation} Joining (\ref{eqr1}), (\ref{eqr2}) and (\ref{eqr3}) we have \[ J(v)=J(w)+J(s u_\varepsilon) \le 0, \] for $\delta$ small, $R$ sufficiently large and $\varepsilon$ sufficiently small. \noindent (2) Let $v\in \Gamma_2$, i.e. $v=w+R u_\varepsilon$ with $\|w\|\le R$; then \begin{eqnarray*} J(v) & = & J(w)+J(R u_\varepsilon) \le c ( m^{-N}+\delta) R^2 +\frac{R^2}{2} \int_{B_{\frac{1}{m}}} | \nabla u_\varepsilon|^2 \,dx \\ & & - \frac{R^{2^\star}}{2^\star} \int_{B_{\frac{1}{m}}} u_\varepsilon^{2^\star} \, dx+ R \| f \| \, \| u_\varepsilon \| <0 \end{eqnarray*} for $R$ sufficiently large. \noindent (3) Let $v\in \Gamma_3=\bar{B}_R\cap H^-_m$. Then \begin{eqnarray*} J(v) & \le& c_1 ( m^{-N}+\delta) \|w\|^2 -\frac{1}{2^\star} \int_{B_{\frac{1}{m}}} ( w+\phi_t(\lambda) )^{2^\star}_+ \,dx \\ & \le& c_1 ( m^{-N}+\delta) \|w\|^2 < \alpha_0 \end{eqnarray*} if $m\ge m_1$ and $\delta>0$ sufficiently small. \noindent ii) For $v \in Q_m$ we have as in Lemma \ref{ma1} inequality (\ref{ri3}), \[ J(v) \le \frac{1}{N} S^{N/2} +c_1 ( m^{-N}+\delta) R^2 +c_2 (\varepsilon m )^{N-2}-c_3 \varepsilon^2-c_4 \varepsilon m^{-N/2} \] If $\varepsilon^\beta=1/m$, with $\beta$ as in Lemma \ref{ma1}, then \[ J(v)\le \frac{S^{N/2}}{N}-c {\varepsilon}^2+c_4 \delta R^2 \] Then there exists $\delta_1$ such that if $0<\delta<\delta_1$, then \[ J(v)\le \frac{S^{N/2}}{N}-\frac{c}{2} {\varepsilon}^2. \] Arguing as in Theorem \ref{t1}, we find a critical point $v_\lambda$ of the functional $J_\lambda$ at a level $c_\lambda\ge\bar{\alpha}>0$ with $\bar{\alpha}$ independent of $\lambda\in(\lambda_k-\delta,\lambda_k)$. It was shown in \cite[Theorem 1.3]{DJ} that in every $\lambda_k, k\ge 1$, starts a bifurcation branch $(\lambda,k)$ emanating from the negative solution and bending to the left (Prop. 4.2) (and thus ``corresponding" to our second solution). We conclude that for $\lambda$ to the left and close to $\lambda_k$ there exist at least three solutions for equation (\ref{p1}), for $t>T_0$ with $T_0=T_0(h)$ sufficiently large. \hfill %$\square$ \begin{thebibliography}{00} \frenchspacing \bibitem{AP} A. Ambrosetti, G. Prodi, {\it {On the inversion of some differentiable mappings with singularities between Banach spaces}}, Ann. Mat. Pura Appl., {\bf 93}, 1972, 231--247. \bibitem{BL} H. Brezis, E. Lieb, {\it A relation between pointwise convergence of functions and convergence of integrals}, Proc. Amer. Math. Soc., {\bf 88}, 1983, 486--490. \bibitem{BN} H. Brezis, L. Nirenberg, {\it Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents}, Comm. Pure Appl. 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Trudinger, {\it Elliptic partial differential equations of second order}, Springer--Verlag, New York, 1993. \bibitem{Ra} P. Rabinowitz, {\it Minimax methods in critical point theory with applications to differential equations}, 65 AMS Conf. Sec. Math, 1986. \bibitem{RS} B. Ruf, B., P.N. Srikanth, {\it Multiplicity results for superlinear elliptic problems with partial interference with spectrum}, J. Math. Anal. App., {\bf 118}, 1986, 15--23. \bibitem{St} M. Struwe, {\it Variational methods}, Springer--Verlag, Berlin, 1996. \end{thebibliography} \noindent\textsc{Marta Calanchi} (e-mail: calanchi@mat.unimi.it, ph. +39.02.50316144)\\ \textsc{Bernhard Ruf} (e-mail: ruf@mat.unimi.it, ph. +39.02.50316157)\\[3pt] Dip. di Matematica, Universit\`{a} degli Studi,\\ Via Saldini 50, 20133 Milano, Italy \end{document}