\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 27, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2003 Southwest Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/27\hfil A nonlinear boundary condition at resonance] {Weak solutions for the $p$-Laplacian with a nonlinear boundary condition at resonance} \author[S. Mart\'\i nez \& J. D. Rossi\hfil EJDE--2003/27\hfilneg] {Sandra Mart\'\i nez \& Julio D. Rossi} \address{Sandra Mart\'\i nez\hfill\break Departamento de Matem\'atica, FCEyN \hfill\break UBA (1428) Buenos Aires, Argentina} \email{smartin@dm.uba.ar} \address{Julio D. Rossi \hfill\break Departamento de Matem\'atica, FCEyN \hfill\break UBA (1428) Buenos Aires, Argentina \hfill\break and \break Facultad de Matematicas, Universidad Catolica. \hfill\break Casilla 306 Correo 22 Santiago, Chile} \email{jrossi@dm.uba.ar, jrossi@mat.puc.cl} \date{} \thanks{Submitted January 15, 2003. Published March 13, 2003.} \subjclass[2000]{35P05, 35J60, 35J55} \keywords{p-Laplacian, nonlinear boundary conditions, resonance} \thanks{Supported by ANPCyT PICT No. 03-00000-00137 and Fundaci\'on Antorchas.} \begin{abstract} We study the existence of weak solutions to the equation $$ \Delta_p u = |u|^{p-2} u+f(x,u) $$ with the nonlinear boundary condition $$ |\nabla u|^{p-2} \frac{\partial u}{\partial\nu} = \lambda |u|^{p-2} u -h(x,u)\,. $$ We assume Landesman-Lazer type conditions and use variational arguments to prove the existence of solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{prop}[theorem]{Proposition} \allowdisplaybreaks \begin{section}{Introduction} This paper shows conditions for the existence of weak solutions to the problem \begin{equation} \label{1.1} \begin{gathered} \Delta_p u = |u|^{p-2} u+f(x,u) \quad \mbox{in } \Omega,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu} = \lambda |u|^{p-2} u-h(x,u) \quad \mbox{on } \partial \Omega. \end{gathered}\end{equation} Here $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, $\Delta_p u= \mathop{\rm div}(|\nabla u|^{p-2} \nabla u)$ is the $p$-Laplacian with $p>1$, and $\frac{\partial}{\partial\nu}$ is the outer normal derivative. We assume that the perturbations $f:\Omega\times\mathbb{R}\to\mathbb{R}$ and $h: \partial\Omega\times\mathbb{R}\to\mathbb{R}$ are bounded Caratheodory functions. For a variational approach, the functional associated to the problem is $$ J_{\lambda}(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^p+\frac{1}{p}\int_{\Omega}|u|^p-\frac{\lambda}{p}\int_{\partial\Omega}| u|^p+\int_{\Omega} F(x,u)+\int_{\partial\Omega}H(x,u), $$ where $F$ and $H$ are primitives of $f$ and $h$ with respect to $u$ respectively. Weak solutions of (\ref{1.1}) are critical points of $J_{\lambda}$ in $W^{1,p} (\Omega)$. In fact if $u \in W^{1,p}(\Omega)$ is a critical point of $J_\lambda$, we have \begin{align*} J'_{\lambda}(u)\cdot v= & \int_{\Omega}|\nabla u|^{p-2} \nabla u \nabla v +\int_{\Omega}|u|^{p-2} u v -\lambda\int_{\partial\Omega}| u|^{p-2}u v \\ & +\int_{\Omega}f(x,u)v+\int_{\partial\Omega}h(x,u)v=0,\quad \forall\ v\in W^{1,p}(\Omega). \end{align*} Let us introduce some notation. We say that $\lambda$ is an eigenvalue for the $p$-Laplacian with a nonlinear boundary condition if the problem \begin{equation}\label{1.2} \begin{gathered} \Delta_p u = |u|^{p-2} u \quad \mbox{in } \Omega,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu} = \lambda |u|^{p-2} u \quad \mbox{on } \partial \Omega. \end{gathered}\end{equation} has non trivial solutions. The set of solutions (called eigenfunctions) for a given $\lambda$ will be denoted by $A_{\lambda}$. Problems of the form \eqref{1.2} appear in a natural way when one considers the Sobolev trace inequality. In fact, the immersion $W^{1,p} (\Omega) \hookrightarrow L^{p} (\partial \Omega)$ is compact, hence there exits a constant $\lambda_1$ such that $$ \lambda_1^{1/p} \| u \|_{L^{p} (\partial \Omega) } \le \|u\|_{W^{1,p} (\Omega)}. $$ The Sobolev trace constant $\lambda_1$ can be characterized as \begin{equation}\label{carac} \lambda_1 = \inf_{u \in W^{1,p} (\Omega)} \Big\{ \int_{\Omega} |\nabla u|^p + |u|^p \, dx \mbox{ such that } \int_{\partial \Omega} |u|^p =1 \Big\}, \end{equation} and is the first eigenvalue of (\ref{1.2}) in the sense that $\lambda_1 \le \lambda$ for any other eigenvalue $\lambda$. The extremals (functions where the constant is attained) are solutions of \eqref{1.2}. The first eigenvalue is simple and isolated with a first eigenfunction that is $C^{\alpha}(\overline{\Omega})$ and strictly positive in $\overline{\Omega}$, see \cite{MR}. In \cite{FBR2} it is proved that there exists a sequence of eigenvalues $\lambda_n$ of \eqref{1.2} such that $\lambda_n\to+\infty$ as $n\to+\infty$. The study of the eigenvalue problem when the nonlinear term is placed in the equation, that is when one considers a quasilinear problem of the form $-\Delta_p u = \lambda |u|^{p-2} u$ with Dirichlet boundary conditions, has received considerable attention, see for example \cite{AH,A,GAP,GP1,Li}, etc. Resonance problems are well known in the literature. For example, for the resonance problem for the $p$-lapacian with Dirichlet boundary conditions see \cite{AG,AO,D-R} and references therein. In problem (\ref{1.1}) we have a perturbation of the eigenvalue problem (\ref{1.2}) given by the two nonlinear terms $f(x,u)$, $h(x,u)$. Following ideas from \cite{D-R}, we prove the following result, that establishes Landesman-Lazer type conditions on the nonlinear perturbation terms in order to have existence of weak solutions for \eqref{1.1}. \begin{theorem}\label{teo3} Let $f^\pm:=\lim_{t\to \pm\infty} f(x,t)$, $h^\pm:=\lim_{t\to \pm\infty} h(x,t)$. Assume that there exists \={f} $\in L^q(\Omega)$ and \={h} $\in L^q(\partial\Omega)$, such that $|f(x,t)|\leq \overline{f}\ \forall (x,t)\in\ \Omega\times\mathbb{R}$ and $|h(x,t)|\leq \overline{h}\ \forall (x,t)\in\ \partial\Omega\times\mathbb{R}$ (where $q=p/p-1$). Also assume that either \begin{equation} \label{LLp} \int_{\{v>0 \cap \Omega\}}f^+v+\int_{\{v>0\cap \partial\Omega\}}h^+v+\int_{\{v<0\cap\Omega\}}f^-v +\int_{\{v<0\cap\partial\Omega\}}h^-v>0 \end{equation} for all $v\in A_{\lambda}\backslash\{0\}$, or \begin{equation} \label{LLm} \int_{\{v>0 \cap \Omega\}}f^+v+\int_{\{v>0\cap \partial\Omega\}}h^+v+\int_{\{v<0\cap\Omega\}}f^-v +\int_{\{v<0\cap\partial\Omega\}}h^-v<0 \end{equation} for all $v\in A_{\lambda}\backslash\{0\}$, then \eqref{1.1} has a weak solution. \end{theorem} Note that when $\lambda$ is not an eigenvalue the hypotheses trivially hold. The integral conditions (of Landesman-Lazer type) that we impose for $f$ and $h$ will be used to prove a Palais-Smale condition for the functional $J_{\lambda}$ associated to the problem (\ref{1.1}). Observe that these conditions involve an integral balance (with the eingenfunctions $v$ as weights) between $f$ and $h$. Hence we allow perturbations both in the equation and in the boundary condition. Let us have a close look at the conditions for the first eigenvalue. As the first eigenvalue is isolated and simple with an eigenfunction that do not change sign in $\Omega$ (we call it $\phi_1$ and assume $\phi_1>0$ in $\bar{\Omega}$), \cite{MR}, the conditions involved in Theorem \ref{teo3} for $\lambda_1$ read as \begin{equation} \label{LLp1} \int_{\Omega}f^+\phi_1+\int_{\partial\Omega} h^+\phi_1>0 \quad \mbox{and} \quad \int_{\Omega}f^-\phi_1+\int_{\partial\Omega}h^-\phi_1 <0 \end{equation} or \begin{equation} \label{LLm1} \int_{\Omega}f^+\phi_1+\int_{\partial\Omega} h^+\phi_1 <0 \quad \mbox{and} \quad \int_{\Omega}f^-\phi_1+\int_{\partial\Omega}h^-\phi_1 >0. \end{equation} For this case, $\lambda=\lambda_1$, we will prove a general result which improve the conditions on $f$ and $h$. In \cite{AG} the resonance problem for the Dirichlet problem was analyzed using bifurcation theory. If we adapt the arguments of \cite{AG} to our situation, using bifurcation techniques to deal with \eqref{1.1}, we can improve the previous result by measuring the speed and the form at which $f$ and $h$ approaches the limits $f^\pm$ and $h^\pm$. To this end, let us suppose that there exists $\alpha$ and $\beta$ such that \begin{gather*} \lim_{s \to +\infty} (f(x,s)-f^+(x)) s^{\alpha}=A_{\alpha}(x),\\ \lim_{s \to -\infty} (f(x,s)-f^-(x)) s^{\beta}= B_{\beta}(x), \quad \mbox{a.e. } x \in \Omega,\\ \lim_{s \to +\infty} (h(x,s)-h^+(x)) s^{\alpha}= \overline{A}_{\alpha}(x), \quad \\ \lim_{s \to -\infty} (h(x,s)-h^-(x)) s^{\beta}= \overline{B}_{\beta}(x), \quad\mbox{a.e, } x \in \partial\Omega. \end{gather*} The limits $A_{\alpha}, \overline{A}_{\alpha}, B_{\beta}$ and $\overline{B}_{\beta}$ are taken in a pointwise sense and dominated by functions in $L^1(\Omega)$ and $L^1(\partial\Omega)$. We consider the conditions: \begin{itemize} \item[$(G^+_{\alpha})$] $\displaystyle \int_{\Omega}f^+\phi_1+\int_{\partial\Omega}h^+\phi_1>0$ or \\ $\displaystyle \int_{\Omega}f^+\phi_1+\int_{\partial\Omega}h^+\phi_1=0$ and $\displaystyle \int_{\Omega} A_{\alpha} (x) \phi_1^{1-\alpha} +\int_{\partial\Omega} \overline{A}_{\alpha} (x) \phi_1^{1-\alpha}>0$ \smallskip \item[$(G^-_{\beta})$] $\displaystyle \int_{\Omega}f^-\phi_1+\int_{\partial\Omega}h^-\phi_1<0$ or \\ $\displaystyle \int_{\Omega}f^-\phi_1+\int_{\partial\Omega}h^-\phi_1=0$ and $\displaystyle \int_{\Omega} B_{\beta} (x) \phi_1^{1-\beta} +\int_{\partial\Omega} \overline{B}_{\beta} (x) \phi_1^{1-\beta} < 0$\smallskip \item[$(G^+_{\beta})$] $\displaystyle \int_{\Omega}f^-\phi_1+\int_{\partial\Omega}h^-\phi_1 > 0$ or\\ $\displaystyle \int_{\Omega}f^-\phi_1+\int_{\partial\Omega}h^-\phi_1=0$ and $\displaystyle \int_{\Omega} B_{\beta} (x) \phi_1^{1-\beta} + \int_{\partial\Omega} \overline{B}_{\beta} (x) \phi_1^{1-\beta}> 0$ \smallskip \item[$(G^-_{\alpha})$] $\displaystyle \int_{\Omega}f^+\phi_1+\int_{\partial\Omega}h^+\phi_1 <0$ or \\ $\displaystyle \int_{\Omega}f^+\phi_1+\int_{\partial\Omega}h^+\phi_1=0$ and $\displaystyle \int_{\Omega} A_{\alpha} (x) \phi_1^{1-\alpha} +\int_{\partial\Omega} \overline{A}_{\alpha} (x)\phi_1^{1-\alpha} <0$. \end{itemize} Where $f^\pm:=\lim_{t\to \pm\infty} f(x,t)$ and $h^\pm:=\lim_{t\to \pm\infty} h(x,t)$. We remark that this set of conditions extend the hypothesis of Theorem \ref{teo3}. \begin{theorem}\label{teo4} Let $f$ and $h$ be such that there exists $f$ in $L^q(\Omega)$ and $h$ in $L^q(\partial\Omega)$, with $|f(x,t)|\leq \overline{f}$ for all $(x,t)\in\ \Omega\times\mathbb{R}$ and $|h(x,t)|\leq \overline{h}$ for all $(x,t)$ in $\partial\Omega\times\mathbb{R}$ (where $q=p/p-1$). If $(G^+_{\alpha})$ and $(G^-_{\beta})$ or $(G^-_{\alpha})$ and $(G^+_{\beta})$ hold then \eqref{1.1} with $\lambda=\lambda_1$ has at least one solution. \end{theorem} We can continue with this procedure and obtain even more general conditions considering the rate of convergence to zero of $(f(x,s) - f^+ (x) )s^\alpha - A_\alpha (x)$, for example. We leave the details to the reader. Also it is possible to consider different rates of convergence, in this case the conditions involve signs of integrals of $A_\alpha$ and $B_\alpha$ separately. In the case $p=2$, we have a linear operator in the Hilbert space $H^1(\Omega)$, so using the Spectral Theorem for compact self-adjoint linear operators and the Fredholm alternative, we have that when $\lambda$ is not an eigenvalue we do not need any additional condition to have solutions for \eqref{1.1}, and if $\lambda$ is an eigenvalue, we need an orthogonality condition. However when dealing with $p\neq 2$ we have to consider the problem in $W^{1,p}(\Omega)$ (which is not Hilbert) and the results is not straightforward. Note that nonlinear boundary conditions have only been considered in recent years. For reference purposes, we cite previous works. For the Laplace operator with nonlinear boundary conditions see for example \cite{ChSF,ChChFS,FdP}. For previous work for the $p$-Laplacian with nonlinear boundary conditions of different types see \cite{CR}, \cite{FBR2}, \cite{P} and \cite{MR}. Also, one is lead to nonlinear boundary conditions in the study of conformal deformations on Riemannian manifolds with boundary, see for example \cite{E2}. \end{section} \begin{section}{Proof of the results} %sec 2. In this section we prove theorems 1.1 and 1.2 that provide existence of solutions to (\ref{1.1}). First, let us prove Theorem \ref{teo3}. We will divide the proof in two steps. Following \cite{D-R}, we first prove a Palais-Smale condition for the functional $J_{\lambda}$ using the conditions of Theorem \ref{teo3}. Then we split the proof of the theorem in two cases, first we deal with $\lambda_k<\lambda<\lambda_{k+1}$, where $\lambda_k$ are the variational eigenvalues of (\ref{1.2}) this allows us to obtain some geometric structure on $J_{\lambda}$ (see \cite{FBR2}), and finally we treat the case where $\lambda=\lambda_k$. In this case we obtain solutions as limit of solutions for a sequence $\lambda_n \to \lambda_k$. We will see that if there is any bifurcation from infinity in $\lambda=\lambda_k$ then the bifurcation is subcritical. This fact provides a priori bounds that allow us to pass to the limit in a sequence of solutions as $\lambda_n \to \lambda_k$. To prove these results we will need some preliminary lemmas (the proofs are straightforward, see \cite{FBR2}). \begin{lemma} \label{lm2.1} Let $A:W^{1,p}(\Omega)\to {W^{1,p}(\Omega)}^*$ be given by $$A(u)\cdot v:=\int_{\Omega}|\nabla u|^{p-2} \nabla u \nabla v+ \int_{\Omega}|u|^{p-2} u v,$$ then $A$ is a continuous, odd, $(p-1)$-homogeneous and continuously invertible. \end{lemma} \begin{lemma} \label{lm2.2} Let $B:W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ be given by $$ B(u)\cdot v:=\int_{\partial\Omega}|u|^{p-2} u v. $$ Then $B$ is a continuous, odd, $(p-1)$-homogeneous and compact. \end{lemma} \begin{lemma} \label{lm2.3} Let $C:W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ be given by $$C(u)\cdot v:= \int_{\Omega}f(x,u)v+\int_{\partial\Omega}h(x,u)v.$$ Then $C$ is continuous and compact and $\|C(u)\|_{W^{1,p}(\Omega)^*}\leq \|\overline{f}\|_{L^q(\Omega)}+K\|\overline{h}\|_{L^q(\partial\Omega)}$. where $K$ is the best constant for the Sobolev trace inequality $W^{1,p}(\Omega) \hookrightarrow L^p(\partial\Omega)$. \end{lemma} With these lemmas we can prove the following theorem. \begin{theorem} \label{thm2.1} Suppose that the hypotheses of Theorem \ref{teo3} are satisfied, then $J_{\lambda}$ satisfies the Palais-Smale condition, that is, for any sequence $\{u_n\}\subset W^{1,p}(\Omega)$ such that $\|J_{\lambda}(u_n)\|_{W^{1,p}(\Omega)}\leq c$ and $J'_{\lambda}(u_n)\to 0$ there exists $u \in W^{1,p}(\Omega)$ such that $u_n\to u$ strongly in $W^{1,p}(\Omega)$. \end{theorem} \begin{proof} Let $\{u_n\}$ be a Palais-Smale sequence. If $u_n$ is bounded then we have that there exists $u \in W^{1,p}(\Omega)$ such that $u_n\rightharpoonup u$ weakly in $W^{1,p}(\Omega)$. Using that $$ A(u_n)-\lambda B(u_n)+C(u_n)=J'_{\lambda}(u_n)\to 0, $$ the compactness of $B$ and $C$, and the continuity of $A^{-1}$ we have that $$u_n\to A^{-1}(\lambda B(u)-C(u)) $$ strongly in $W^{1,p}(\Omega)$. Hence if we prove that Palais-Smale sequences are bounded, the result follows. To see this, let us argue by contradiction. Assume that $u_n$ is a Palais-Smale sequence and that $\|u_n\|_{W^{1,p}(\Omega)}\to \infty$. Let $$ v_n:=\frac{u_n}{\|u_n\|_{W^{1,p}(\Omega)}}$$ then there exists $v$ such that $v_n\rightharpoonup v$ in $W^{1,p}(\Omega)$ and $v_n\to v$ in $L^p(\partial\Omega)$. We have, \begin{equation} \label{ps} \frac{J'_{\lambda}(u_n)}{\|u_n\|_{W^{1,p}(\Omega)}^{p-1}}=A(v_n)-\lambda B(v_n)+\frac{C(u_n)}{\|u_n\|_{W^{1,p}(\Omega)}^{p-1}}. \end{equation} Using compactness of $B$, continuity of $A^{-1}$ and the fact that $$ \frac{C(u_n)}{\|u_n\|_{W^{1,p}(\Omega)}^{p-1}}\to 0 $$ we have that $v_n\to A^{-1}(\lambda B(v))$ in $W^{1,p}(\Omega)$. Hence $v_n\to v$ in $W^{1,p}(\Omega)$ and then $A(v)-\lambda B(v)=0$ with $\|v\|_{W^{1,p}(\Omega)}=1$. That means that $v \in A_{\lambda}\backslash\{0\}$. Observe that, for a.e. $x\in\{v(x)>0\}$, we have $u_n(x)\to +\infty$ so, $$ \lim_{n\to \infty} f(x,u_n(x)) v_n(x)+h(x,u_n(x))v_n(x)=f^+(x) v(x)+h^+(x) v(x),$$ and \begin{align*} &\lim_{n\to \infty} \frac {F(x,u_n(x))}{ \|u_n\|_{W^{1,p}(\Omega)}}+ \frac {H(x,u_n(x))}{ \|u_n\|_{W^{1,p}(\Omega)}} \\ & = \lim_{n\to \infty}v_n(x) \frac{1}{u_n(x)} \int_{0}^{u_n(x)} f(t,u_n(t))+v_n(x) \frac{1}{u_n(x)} \int_{0}^{u_n(x)} h(t,u_n(t))\\ &=v(x)f^+(x)+v(x)h^+(x). \end{align*} In a similar way we obtain that, for a.e. $x\in \{x:v(x)<0\}$, we have $$\lim_{n\to \infty} f(x,u_n(x)) v_n(x)+h(x,u_n(x))v_n(x)=f^-(x) v(x)+h^-(x) v(x),$$ and therefore $$ \lim_{n\to \infty} \frac {F(x,u_n(x))}{ \|u_n\|_{W^{1,p}(\Omega)}}+ \frac {H(x,u_n(x))}{ \|u_n\|_{W^{1,p}(\Omega)}}=v(x)f^-(x)+v(x)h^-(x). $$ On the other hand, we have \begin{align*} &pJ_{\lambda}(u_n)-J'_{\lambda}(u_n)\cdot u_n\\ &=p\int_{\Omega}F(x,u_n(x))+p\int_{\partial\Omega}H(x,u_n(x)) -\int_{\Omega}f(x,u_n(x))u_n-\int_{\partial\Omega}h(x,u_n(x))u_n. \end{align*} Then \begin{align*} &p\frac{J_{\lambda}(u_n)}{ \|u_n\|_{W^{1,p}(\Omega)}}-J'_{\lambda}(u_n)\cdot v_n\\ &=p\int_{\Omega}\frac{F(x,u_n(x))}{ \|u_n\|_{W^{1,p}(\Omega)}}+p\int_{\partial\Omega}\frac{H(x,u_n(x))}{ \|u_n\|_{W^{1,p}(\Omega)}} -\int_{\Omega}f(x,u_n(x))v_n-\int_{\partial\Omega}h(x,u_n(x))v_n. \end{align*} The left hand side approaches $0$ as $n\to \infty$. Hence $$ 0=(p-1)\Big[\int_{\{v>0 \cap \Omega\}}f^+v+\int_{\{v>0\cap \partial\Omega\}}h^+v+\int_{\{v<0\cap\Omega\}}f^-v+ \int_{\{v<0\cap\partial\Omega\}}h^-v\Big] $$ which contradicts the hypothesis on $f$ and $h$ in Theorem \ref{teo3}. \end{proof} Now that we have proved the Palais-Smale condition, we can state a deformation theorem that will be used later to show that $J_{\lambda}$ has critical points (see \cite{st}). \begin{theorem}\label{teodef} Suppose that $J_{\lambda}$ satisfies the Palais-Smale condition. Let $\beta\in\mathbb{R}$ be a regular value of $J_{\lambda}$ and let $\bar{\epsilon}>0$. Then there exists $\epsilon\in(0,\bar{\epsilon})$ and a continuous one-parameter family of homeomorphisms, $\Phi: W^{1,p}(\Omega)\times[0,1]\to W^{1,p}(\Omega)$ with the following properties: \begin{enumerate} \item $\Phi(u,t)=u$ if $t=0$ or if $|J_{\lambda}-\beta|\geq\bar{\epsilon}$. \item $J_{\lambda}(\Phi(u,t))$ is non decreasing in $t$ for any $u\in W^{1,p}(\Omega)$. \item If $J_{\lambda}(u)\leq \beta+\epsilon$ then $J_{\lambda}(\Phi(u,1))\leq\beta-\epsilon$. \end{enumerate} \end{theorem} We now use a variational characterization for a sequence of eigenvalues for the problem (\ref{1.2}). Indeed, solutions of (\ref{1.2}) we can understood as critical points of the associated energy functional $$ I (u) =\int_{\Omega } |\nabla u|^p + \int_{\Omega}|u|^p , $$ under the constraint $u \in M$, where $M=\{u \in W^{1,p}(\Omega):\|u\|_{L^p(\partial\Omega)}=1\}$. We can find a sequence of variational eigenvalues with the characterization, $$ \lambda_k:=\inf_{A\in C_k} \sup_{u \in A} I(u),$$ where $$ C_k:=\{ A\subset M: \mbox{there exists } h:S^{k-1}\to A \mbox{ continuous, odd and surjective } \}. $$ To prove that these $\lambda_k$ are critical values one first proves a Palais-Smale condition for the functional. Next, using a deformation argument, we prove that $\lambda_k$ is an eigenvalue (see \cite{FBR2} for the details), but it is not known if this sequence contains all the eigenvalues. As we mentioned before, we divide the proof in two cases, $\lambda_k< \lambda< \lambda_{k+1}$ and $\lambda=\lambda_k$. \noindent {\bf Case $\lambda_k< \lambda< \lambda_{k+1}$.} Let $A\in C_k$ such that $ \sup_{u\in A} I(u)=m \in (\lambda_k,\lambda)$ (here we are using the definition of $\lambda_k)$. Then we have, for $u\in\ A,\ t>0$, that \begin{align*} J_{\lambda}(tu)&= \frac{t^p}{p}[\|u\|^p_{W^{1,p}(\Omega)}-\lambda]+\int_{\Omega} F(x,tu)+\int_{\partial\Omega} H(x,tu)\\ &\leq \frac{t^p}{p} (m-\lambda)+\Big|\int_{\Omega} F(x,tu)\Big| +\Big|\int_{\partial\Omega} H(x,tu)\Big|\\ &\leq \frac{t^p}{p} (m-\lambda) t\Big(\int_{\Omega} |u|^p\Big)^{1/p}\Big(\int_{\Omega} |\overline{f}|^q\Big)^{1/q}+t\Big(\int_{\partial\Omega}|u|^p\Big)^{1/p} \Big(\int_{\partial\Omega}|\overline{h}|^q\Big)^{1/q}\\ &\leq \frac{t^p}{p} (m-\lambda)+t \left(m \|\overline{f}\|_{L^q(\Omega)} + \|\overline{ h}\|_{L^q(\partial\Omega)}\right). \end{align*} Let $$\xi_{k+1}=\Big\{u\in\ W^{1,p}(\Omega) : \int_{\Omega} |\nabla u|^p+\int_{\Omega} |u|^p\geq \lambda_{k+1} \int_{\partial\Omega} |u|^p\Big\}. $$ If $u\in \xi_{k+1}$ then, \begin{align*} J_{\lambda}(u)&= \frac{1}{p} \big[\int_{\Omega} |\nabla u|^p+\int_{\Omega}|u|^p\big] -\frac{\lambda}{p}\int_{\partial\Omega}|u|^p +\int_{\Omega}F(x,u)+\int_{\partial\Omega}H(x,u)\\ &\geq \frac{1}{p}\|u\|^p_{W^{1,p}(\Omega)} \big[1-\frac{\lambda}{\lambda_{k+1}}\big]+ \int_{\Omega}F(x,u)+\int_{\partial\Omega}H(x,u)\\ & \geq \frac{1}{p}\|u\|^p_{W^{1,p}(\Omega)} \big[1-\frac{\lambda}{\lambda_{k+1}}\big] -\|u\|_{W^{1,p}(\Omega)}\|\overline{f}\|_{L^q(\Omega)}\\ &\quad -K\|u\|_{W^{1,p}(\Omega))}\|\overline{h}\|_{L^q(\partial\Omega)}. \end{align*} This proves the coercitivity of $J_{\lambda}$ in $\xi_{k+1}$, then there exists $\alpha$ such that, $$ \alpha:= \inf_{u \in \xi_{k+1}} J_{\lambda}(u). $$ On the other hand we have, for $u\in A$, $$ J_{\lambda}(tu)\leq \frac{t^p}{p} (m-\lambda)+t \left(m\|\overline{f}\|_{L^q(\Omega)}+ \|\overline{h}\|_{L^q(\partial\Omega)}\right), $$ where $m-\lambda<0$. Then for all $u\in\ A$, as $t\to +\infty$ $J_{\lambda}(tu)\to -\infty$. Hence there exists $T>0$ such that \begin{equation}\label{max} \max_{u \in A, t\geq T} J_{\lambda}(tu)=\gamma<\alpha. \end{equation} Let $TA:=\{tu:\ u\in \ A,\ t\geq T\}$ and $$ \chi:=\{h\in\ C(B_k(0,1),W^{1,p}(\Omega)):\ h|_{S^{k-1}}\mbox{is odd into } TA\}. $$ Let us show that $\chi$ is nonempty. By the definition of $C_k$, there exists continuous function $h:S^{k-1}\to A$ odd and surjective. Let us define $\overline{h}:B_k\to W^{1,p}(\Omega)$ as $\overline{h}(ts)=tTh(s) \ s\in\ S^{k-1},\ t\in\ [0,1]$. Clearly $\overline{h}\in \chi$. Next, let we prove that if $h\in \chi$ then $h(B_k)\cap\xi_{k+1}\neq \emptyset$. If there exists any $u\in h(B_k)$ such that $\int_{\partial\Omega}|u|^p=0$ then $u \in \xi_{k+1}$. Suppose now that $\int_{\partial\Omega}|u|^p\neq 0$ for all $u\in h(B_k)$, and let us consider $$ \widetilde{h}(x_1,\dots ,x_{k+1})= \begin{cases} \pi h(x_1,\dots ,x_k) & x_{k+1}\geq 0\\ -\pi h(-x_1,\dots ,-x_k) & x_{k+1}< 0, \end{cases} $$ where $\pi u=u/ \|u\|_{L^p(\partial\Omega)}$. Then, if $x_{k+1}\geq 0$, $$ \widetilde{h}(x_1,\dots ,x_{k+1})=\pi (-h(-x_1,\dots ,-x_k))=-\pi h(-x_1,\dots ,-x_k) $$ and hence $$ \widetilde{h}(-x_1,\dots ,-x_{k+1})=-\pi h(x_1,\dots ,x_k) =-\widetilde{h}(x_1,\dots ,x_{k+1}). $$ In an analogous way for $x_{k+1}< 0$, we have $$ \widetilde{h}(x_1,\dots ,x_{k+1})=-\widetilde{h}(-x_1,\dots ,-x_{k+1}), $$ then $\widetilde{h}$ is odd. Hence $\widetilde{h}(S^{k}) \in C^{k+1}$. On the other hand, we have, $$ \lambda_{k+1}=\inf_{A \in C^{k+1}} \sup_{u \in A } I(u), $$ then $$ \lambda_{k+1}\leq \sup_{u \in \widetilde{h}(S^k)} I(u). $$ Hence, for some $u \in \widetilde{h}(S^k)$, that is, for some $x\in S^k$ such that $u=\widetilde{h}(x)$ we have $\lambda_{k+1}\leq I(u)$. This implies that $\widetilde{h}(x) \in \xi_{k+1}$. Using the definition of $\widetilde{h} $ we obtain that $h(x) \in \xi_{k+1}$. Then $h(B_k) \cap \xi_{k+1}\neq \emptyset $. \begin{theorem} The value $$c:=\inf_{h \in \chi} \sup_{x \in B_k} J_{\lambda} h(x),$$ is a critical value for $J_{\lambda}$, with $c\geq\alpha$. \end{theorem} \begin{proof} For each $h \in \chi$, there exists $x\in B_k$ such that $h(x) \in \xi_{k+1}$, then $J_{\lambda} (h(x))\geq \alpha$. Hence $$ \sup_{x \in B_k} J_{\lambda}(h(x)) \geq \alpha \quad \forall \ h \in \chi. $$ Therefore, $c\geq \alpha>\gamma$, where $\gamma$ is given by \eqref{max}. Let us argue by contradiction. Suppose that $c$ is a regular value, then using the deformation Theorem \ref{teodef}, with $\beta=c$ and $\bar{\epsilon}< c-\gamma$, we have that there exists a deformation $\Phi(u,t)$ that verifies the usual properties. If $u \in TA$ then, $$ J_{\lambda}(u)\leq \gamma<\beta-\bar{\epsilon}, $$ then by one of the properties of the deformation lemma we have $\Phi(u,t)=u$. By the definition of $c$, there exists $h\in \chi$ such that, \begin{equation} \label{j} \sup_{x \in B_k} J_{\lambda} (h(x))\leq c+\epsilon. \end{equation} Let $\tilde{h}(\cdot):=\Phi(h(\cdot),1)$, if $x \in S^{k-1}$ we have that $h(x) \in TA$ , then $ \tilde{h} (x)=\Phi(h(x),1)=h(x)$ and hence $\tilde{h}|_{S^{k-1}}=h|_{S^{k-1}}$. We also have $\tilde{h}(-x)=\Phi(h(-x),1)=\Phi(-h(x),1)=-\tilde{h}(x)$. We obtain that $\tilde{h} \in \chi$. Using (\ref{j}) and the deformation theorem we have $$\sup_{x \in B_k} J_{\lambda}(\tilde{h}(x))=\sup_{x \in B_k} J_{\lambda}(\Phi(h(x),1)) \leq c-\epsilon,$$ a contradiction that proves that $c$ is a critical value. \end{proof} \noindent{\bf Case $\lambda=\lambda_k$.} We will prove the result under condition $(LL)^+_{\lambda_k}$, the case where $(LL)^-_{\lambda_k}$ holds is completely analogous. \begin{lemma} If $(LL)^+_{\lambda_k}$ is satisfied, then there exists $\delta>0$ such that $(LL)^+_{\mu}$ is satisfied for all $\mu\in (\lambda_k-\delta,\lambda_k+\delta)$. \end{lemma} \begin{proof} Arguing by contradiction, let us assume that there exists $\mu_n\to \lambda_k$ and corresponding eigenfunctions $\{v_n\}$, $\|v_n\|_{W^{1,p}(\Omega)}=1$, such that \begin{equation}\label{auto} \int_{\Omega} |\nabla v_n|^{p-2}\nabla v_n \nabla w +\int_{\Omega} |v_n|^{p-2}v_n w=\mu_n \int_{\partial\Omega} |v_n|^{p-2}v_n\ \ \ \forall w \in W^{1,p}(\Omega) \end{equation} and \begin{equation}\label{formul} \int_{\{v_n>0 \cap \Omega\}}f^+v_n+\int_{\{v_n>0\cap \partial\Omega\}}h^+v_n+\int_{\{v_n<0\cap\Omega\}}f^-v_n +\int_{\{v_n<0\cap\partial\Omega\}}h^-v_n\leq 0, \end{equation} for all $n$. Then, since $\{v_n\}$ is bounded, there exists $v \in W^{1,p}(\Omega)$ such that $v_n\to v $ in $L^p(\partial\Omega)$. Taking $$ \phi_n(w)=\mu_n\int_{\partial\Omega} |v_n|^{p-2} v_n w \quad\mbox{and}\quad \phi(w)=\lambda_k\int_{\partial\Omega} |v|^{p-2} v w, $$ we have that $\phi_n \to \phi$ in $(W^{1,p}(\Omega))^*$. Using the continuity of $A^{-1}$, we have that $v_n\to v$ in $W^{1,p}(\Omega)$. Then, taking limits in (\ref{auto}) and (\ref{formul}) we have $$ \int_{\Omega} |\nabla v|^{p-2}\nabla v \nabla w +\int_{\Omega} |v|^{p-2}v w=\lambda_k \int_{\partial\Omega} |v|^{p-2}v,\ \ \ \forall w \in W^{1,p}(\Omega), $$ and $$ \int_{\{v>0 \cap \Omega\}}f^+v+\int_{\{v>0\cap \partial\Omega\}}h^+v+\int_{\{v<0\cap\Omega\}}f^-v +\int_{\{v<0\cap\partial\Omega\}}h^-v\leq 0. $$ Which contradicts the fact that $(LL)^+_{\lambda_k}$ is satisfied. \end{proof} Now we assume that $\lambda_{k-1}\leq \lambda_k-\delta$ and let $\{\mu_n\} \subset (\lambda_k-\delta, \lambda_k)$ be an increasing sequence such that $\mu_n\to \lambda_k$. We will construct a decreasing sequence $\{c_n\}$ of critical values corresponding to $J_{\mu_n}$, and then we will see that the sequence corresponding to the critical points $\{u_n\}$ is bounded and converges to a certain $u$ that is a critical point for $J_{\lambda_k}$. \begin{lemma} \label{lm2.5} There exists a decreasing sequence of critical values, $\{c_n\}$ associated with the functional $J_{\mu_n}$. \end{lemma} \begin{proof} Let $A \in C^{k-1}$, $T_1>0, \xi_{k}$ and $\chi_1$ as in the first part ($\lambda_k<\lambda<\lambda_{k+1}$) such that, $$ c_1:=\inf_{h\in \chi_1}\sup_{x\in B_{k-1}} J_{\mu_1}(h(x)) $$ is a critical value for $J_{\mu_1}$. To define $c_2$, let us chose the same $A$ and $\xi_k$, but we take $T_2>T_1$ that provides the correspondent $\chi_2$. Then $T_2A\subset T_1A$, $\chi_2\subset \chi_1$ and, $$ \inf_{h\in \chi_2}\sup_{x\in B_{k-1}} J_{\mu_1}(h(x))\geq \inf_{h\in \chi_1}\sup_{x\in B_{k-1}} J_{\mu_1}(h(x))=c_1. $$ Let $$h_2(x):=\begin{cases} h_1(2x) & |x|\leq \frac{1}{2},\\ h_1\big(\frac{x}{|x|}\big) [1+2(|x|-\frac{1}{2} )T_2] & |x|>\frac{1}{2}. \end{cases} $$ For $|x|\geq 1/2$, $h_2(x)\in T_1A$; therefore, $$ J_{\mu_1}(h_2(x))\leq \gamma<\alpha\leq J_{\mu_1}(u),\quad \forall u\in \xi_{k+1}. $$ Then there exists $y \in B_{k}$ such that $h_2(y) \in \xi_{k+1}$ and $$ J_{\mu_1}(h_2(x))\leq \gamma<\alpha\leq J_{\mu_1}(h_2(y)). $$ That is, for all $x$ with $|x|\geq 1/2$ there exists $y \in B_{k}$ such that $J_{\mu_1}(h_2(x))\lambda_k$, that is the only possible bifurcation from infinity at $\lambda=\lambda_k$ is subcritical. \begin{lemma}\label{ap} If $\|u_n\|_{W^{1,p}(\Omega)}\to\infty$, then there exists $v \in A_{\lambda_k}\setminus \{0\}$ such that $$\frac{u_n}{\|u_n\|_{W^{1,p}(\Omega)}}\to v.$$ \end{lemma} \begin{proof} Let $v_n:=u_n/ \|u_n\|_{W^{1,p}(\Omega)}$. Then $v_n\rightharpoonup v$. Using that \begin{equation} \label{a} A(v_n)-\mu_n B(v_n)-\frac{C(u_n)}{\|u_n\|^{p-1}}=0, \end{equation} the compactness of $B$ and the continuity of $A^{-1}$, we have $v_n \to A^{-1}(\lambda_k B(v))$. Then $v_n\to v$, with $\|v\|_{W^{1,p}(\Omega)}=1$. Taking limits in (\ref{a}) we have $A(v)=\lambda_k B(v)$, then $v \in A_{\lambda_k} \setminus\{0\}$. \end{proof} Making similar calculations to those in the proof of Theorem 2.1, we get \begin{align*} pc_n&=p J_{\mu_n}(u_n)-J'_{\mu_n}(u_n)\cdot u_n\\ &=p\int_{\Omega} F(x,u_n)+p\int_{\partial\Omega} H(x,u_n) -\int_{\Omega} f(x,u_n)u_n-\int_{\partial\Omega} h(x,u_n)u_n. \end{align*} Then \begin{align*} &\lim_{n\to\infty}p\int_{\Omega} \frac{F(x,u_n)}{\|u_n\|_{W^{1,p}(\Omega)}}+ p\int_{\partial\Omega} \frac{H(x,u_n)}{\|u_n\|_{W^{1,p}(\Omega)}} -\int_{\Omega} f(x,u_n)v_n-\int_{\partial\Omega} h(x,u_n)v_n\\ &=(p-1)\Big(\int_{\{v>0 \cap \Omega\}}f^+v+\int_{\{v>0\cap \partial\Omega\}}h^+v+\int_{\{v<0\cap\Omega\}}f^-v+ \int_{\{v<0\cap\partial\Omega\}}h^-v\Big)\\ &> 0. \end{align*} Then, $$\lim_{n\to\infty}\frac{pc_n}{\|u_n\|_{W^{1,p}(\Omega)}}>0,$$ which contradicts the fact that $\{c_n\}$ is bounded from above. Then we have that $\{u_n\}$ is bounded. Hence there exists $u \in W^{1,p}(\Omega)$ such that $u_n\rightharpoonup u$ weak in $W^{1,p}(\Omega)$, using the compactness of $B$ and $C$ and the continuity of $A^{-1}$ we have $u_n\to u$ strong in $W^{1,p}(\Omega)$. \noindent{\bf Case $\lambda= \lambda_1$} This corresponds to Theorem \ref{teo4}. In this theorem we improve the conditions on $f$ and $h$ for the case where $\lambda=\lambda_1$. We use ideas from \cite{AG}, but first we find some estimates. \begin{lemma} Let $u\in C^{\alpha}(\Omega)$ be a solution of \eqref{1.1} strictly positive in $\overline{\Omega}$. Then $$ -\frac{\displaystyle\int_{\partial\Omega} h(x,u) \frac{\phi_1^p}{|u|^{p-2}u}+\int_\Omega f(x,u) \frac{\phi_1^p}{|u|^{p-2}u}}{\displaystyle\int_{\partial\Omega} \phi_1^p} \leq \lambda_1-\lambda\leq -\frac{\displaystyle \int_{\partial\Omega} h(x,u) u+ \int_\Omega f(x,u) u}{\displaystyle \int_{\partial\Omega} |u|^p}.$$ \end{lemma} \begin{proof} In the weak form with $v=u$, we have \begin{align*} -\int_{\partial\Omega} g(x,u) u-\int_\Omega f(x,u) u &=\int_\Omega |\nabla u|^p+ \int_\Omega |u|^p -\lambda \int_{\partial\Omega} |u|^p \\ &\geq (\lambda_1-\lambda) \int_{\partial\Omega} |u|^p, \end{align*} then we get the second inequality. If we take $ v=\phi_1^p / (|u|^{p-2}u)$ we have, \begin{align*} &-\int_{\partial\Omega} h(x,u) \frac{{\phi_1}^p}{|u|^{p-2}u}- \int_\Omega f(x,u) \frac{{\phi_1}^p} {|u|^{p-2}u}-(\lambda_1-\lambda)\int_{\partial\Omega} {\phi_1}^p \\ &= \int_\Omega |\nabla u|^{p-2} \nabla u \nabla\big(\frac{\phi_1^p}{|u|^{p-2}u}\big) + \int_\Omega |u|^{p-2}u \frac{\phi_1^p}{|u|^{p-2}u} -\int_{\Omega} |\nabla \phi_1|^p- \int_{\partial\Omega} |\phi_1|^p \\ &= \int_\Omega p |\nabla u|^{p-2} \frac{{\phi_1}^{p-1}}{|u|^{p-2}u} \nabla u \nabla \phi_1-\int_\Omega(p-1) \frac{{\phi_1}^p}{|u|^p}|\nabla u|^p- \int_{\Omega} |\nabla \phi_1|^p \\ &\leq \int_\Omega p \frac{{\phi_1}^{p-1}}{|u|^{p-1}} |\nabla u|^{p-1} |\nabla \phi_1|-\int_\Omega(p-1) \frac{{\phi_1}^p}{|u|^p}|\nabla u|^p-\int_{\Omega} |\nabla \phi_1|^p. \end{align*} Using that $$p t^{p-1}s-(p-1)t^p-s^p \leq 0,\ \ \ \forall t,s \geq 0$$ with $ t=\frac{\phi_1}{|u|} |\nabla u|$ and $ s=|\nabla \phi_1|$ we have that $$-\int_{\partial\Omega} h(x,u) \frac{\phi_1}{|u|^{p-2}u}-\int_\Omega f(x,u) \frac{\phi_1}{|u|^{p-2}u} -(\lambda_1-\lambda)\int_{\partial\Omega} {\phi_1}^p\leq 0,$$ the result follows. \end{proof} Now, let us proceed with the proof of the main theorem. \begin{proof}[Proof of Theorem \ref{teo4}] Let us suppose that $f$ and $h$ satisfy conditions $(G^-_{\alpha})$ and $(G^+_{\beta})$. We will prove that there exists $(\lambda_n,u_n)$ solutions of problem (\ref{1.1}) with $\lambda_n \to \lambda_1$ such that $\|u_n\|_{W^{1,p}(\Omega)}\leq K$. This will follows from the fact that any possible bifurcation from infinity must be subcritical. Let $\lambda_n \searrow \lambda_1$, and $u_n$ be the solutions of (\ref{1.1}). Remark that Theorem \ref{teo3} shows the existence of $u_n$ for every $\lambda_n$ close but not equal to $\lambda_1$ (as $\lambda_1$ is isolated the conditions on $f$ and $h$ of Theorem \ref{teo3} are trivially verified for any $\lambda_n$ close to $\lambda_1$). Suppose that ${\|u_n\|_{W^{1,p}(\Omega)}}\to \infty$. If $u_n/\|u_n\|_{W^{1,p}(\Omega)} \to \phi_1$ and $$\int_{\Omega} f^+ \phi_1+\int_{\partial\Omega} h^+ \phi_1<0$$ then we arrive to a contradiction. Otherwise, if $u_n/\|u_n\|_{W^{1,p}(\Omega)}\to -\phi_1$ and $$ \int_{\Omega} f^- \phi_1+\int_{\partial\Omega} h^- \phi_1<0 $$ we also arrive to a contradiction. Hence in both cases any bifurcation from infinity must be subcritical. Hence $\{u_n\}$ is bounded (see \cite{AG} for the details). We have to consider only the case where \begin{equation} \label{cero} \begin{gathered} \int_{\Omega} f^+ \phi_1+\int_{\partial\Omega} h^+ \phi_1=0,\\ \int_{\Omega} f^- \phi_1+\int_{\partial\Omega} h^-\phi_1=0, \end{gathered} \end{equation} and \begin{gather*} \int_{\partial\Omega} \overline{A}_{\alpha} \phi_1^{1-\alpha} +\int_{\Omega} A_{\alpha} \phi_1^{1-\alpha}<0,\\ \int_{\partial\Omega} \overline{B}_{\alpha} \phi_1^{1-\alpha} +\int_{\Omega} B_{\alpha} \phi_1^{1-\alpha}>0. \end{gather*} Let us assume by contradiction that $\|u_n\|_{W^{1,p}(\Omega)}\to \infty$. Then by Lemma \ref{ap}, $$ \frac{u_n} {\|u_n\|_{W^{1,p}(\Omega)}}\to \pm \phi_1. $$ The convergence is uniform by regularity results that show that $ u_n \in C^\alpha(\Omega)$, see \cite{L}. Using the previous lemma, $$ 0>(\lambda_1-\lambda_n) \int_{\partial\Omega} \phi_1^p\geq -\int_{\partial\Omega} h(x,u_n) \frac{\phi_1^p} {|u_n|^{p-2} u_n} -\int_\Omega f(x,u_n) \frac{\phi_1^p}{|u_n|^{p-2}u_n}. $$ Using (\ref{cero}), \begin{equation}\label{ultima} \begin{aligned} 0< & \int_{\partial\Omega} (h(x,u_n) \phi_1^{p-1} \frac{\|u_n\|^{p-1}}{|u_n|^{p-2} u_n}- h^+(x)) \phi_1\\ &+ \int_\Omega (f(x,u_n)\phi_1^{p-1} \frac{\|u_n\|^{p-1}}{|u_n|^{p-2}u_n} -f^+(x) ) \phi_1 \\ = & \int_{\partial\Omega} (h(x,u_n)-h^+(x)) \phi_1^{p-1} \frac{ \|u_n\|^{p-1}}{|u_n|^{p-2}u_n} \phi_1\\ &\quad - \int_{\partial\Omega} h^+(x) \phi_1(1-\phi_1^{p-1} \frac{\|u_n\|^{p-1}}{|u_n|^{p-2} u_n}) \\ & \quad +\int_{\Omega} (f(x,u_n)-f^+(x)) \phi_1^{p-1} \frac{\|u_n\|^{p-1}}{|u_n|^{p-2}u_n} \phi_1\\ &\quad -\int_{\Omega} f^+(x) \phi_1(1- \phi_1^{p-1} \frac{\|u_n\|^{p-1}}{|u_n|^{p-2}u_n}). \end{aligned} \end{equation} If $u_n/ \|u_n\|_{W^{1,p}(\Omega)}\to \phi_1$, using our hypothesis on the dominated convergence of $(h(x,u_n)-h^+(x)) u_n^{\alpha}$ by a function in $L^1(\partial\Omega)$ and the uniform convergence of $u_n/\|u_n\|_{W^{1,p}(\Omega)}$ to $\phi_1$, we have the hypotheses of the Lebesgue's Dominated Convergence Theorem. The second term also verifies these hypotheses. Then using our hypothesis over $f$ and $h$, and taking the limit we have \begin{align*} &\lim_{n\to\infty}\int_{\partial\Omega} (h(x,u_n)-h^+(x))\|u_n\|^{\alpha}\phi_1^{p} \frac{\|u_n\|^{p-1}}{|u_n|^{p-2} }\\ &+\int_{\Omega} (f(x,u_n)-f^+(x)) \|u_n\|^{\alpha}\phi_1^{p} \frac{\|u_n\|^{p-1}}{|u_n|^{p-2} u_n} \\ & = \int_{\partial\Omega} \overline{A}_{\alpha} \phi_1^{1-\alpha} +\int_{\Omega} A_{\alpha} \phi_1^{1-\alpha} < 0. \end{align*} Therefore, for $n$ large enough \begin{align*} &\int_{\partial\Omega} (h(x,u_n)-h^+(x))\|u_n\|^{\alpha}\phi_1^{p} \frac{\|u_n\|^{p-1}}{|u_n|^{p-2} }\\ &+\int_{\Omega} (f(x,u_n)-f^+(x)) \|u_n\|^{\alpha}\phi_1^{p} \frac{\|u_n\|^{p-1}}{|u_n|^{p-2} u_n} 0, $$ and proceeding as before we arrive to a contradiction. Hence $\{u_n\}$ must be bounded. If $f$ and $h$ satisfy condition $(G^+_{\alpha})$ and $(G^-_{\beta})$, using the other inequality we prove that if we take $(\lambda_n,u_n)$ solutions of (\ref{1.1}) with $\lambda_n\nearrow\lambda_1$ then $\{u_n\}$ must be bounded. Using the same argument as in the previous theorem we see that there exists $u \in W^{1,p}(\Omega)$ such that $u_n\to u$ and $u$ is a solution for (\ref{1.1}) with $\lambda=\lambda_1$. This completes the proof. \end{proof} We can observe that in the proof of the previous theorem we prove that if $f$ and $h$ satisfy the condition $(G^-_{\alpha})$ and $(G^+_{\beta})$ then any bifurcation from infinity must be subcritical, and in the second case any bifurcation from infinity must be supercritical. \end{section} \smallskip \noindent\textbf{Acknowledgements.} We want to thank Professors: D. Arcoya, J. Garcia-Azorero and I. Peral for their suggestions and interesting discussions. \begin{thebibliography}{99} \frenchspacing \bibitem{AH} W. Allegretto and Y. X. 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