\documentstyle[twoside,amssymb]{article} % amssymb is used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Existence and multiplicity of solutions \hfil EJDE--1998/10}% {EJDE--1998/10\hfil Klaus Pf\/l\"uger \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.\ {\bf 1998}(1998), No.~10, pp. 1--13. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \vspace{\bigskipamount} \\ Existence and multiplicity of solutions to a $p$-Laplacian equation with nonlinear boundary condition \thanks{ {\em 1991 Mathematics Subject Classifications:} 35J65, 35J20. \hfil\break\indent {\em Key words and phrases:} p-Laplacian, nonlinear boundary condition, variational methods, \hfil\break\indent unbounded domain, weighted function space. \hfil\break\indent \copyright 1998 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted March 5, 1998. Published April 10, 1998.} } \date{} \author{Klaus Pf\/l\"uger} % Pflueger \maketitle \begin{abstract} We study the nonlinear elliptic boundary value problem \begin{eqnarray*} & A u = f(x,u) \quad \mbox{in }\Omega\,,\\ & Bu = g(x,u)\quad \mbox{on }\partial \Omega\,, \end{eqnarray*} where $A$ is an operator of $p-$Laplacian type, $\Omega$ is an unbounded domain in ${\Bbb R}^N$ with non-compact boundary, and $f$ and $g$ are subcritical nonlinearities. We show existence of a nontrivial nonnegative weak solution when both $f$ and $g$ are superlinear. Also we show existence of at least two nonnegative solutions when one of the two functions $f$, $g$ is sublinear and the other one superlinear. The proofs are based on variational methods applied to weighted function spaces. \end{abstract} \newtheorem{theo}{Theorem} \newtheorem{lemma}{Lemma} \newcommand{\bewende}{\hfill\mbox{$\Box$}\protect} \newcommand{\R}{{\Bbb R}} \newcommand{\vi}{\varphi} \newcommand{\eps}{\varepsilon} \newcommand{\n}{\mbox{\sf n}} \newcommand{\dn}{\partial_{\n}} \newcommand{\diver}{\mathop{\rm div}} \section{Introduction} The objective of this paper is to study the nonlinear elliptic boundary value problem \begin{eqnarray} &-\diver (a(x) |\nabla u|^{p-2} \nabla u) = f(x,u) \quad \mbox{ in } \quad \Omega \subset \R^N, & \label{p1} \\ &\n \cdot a(x) |\nabla u|^{p-2} \nabla u + b(x) |u|^{p-2}u = g(x,u) \quad \mbox{ on } \quad \Gamma = \partial \Omega, & \label{p2} \end{eqnarray} where $\Omega$ is an unbounded domain with noncompact, smooth boundary $\Gamma$ (for example a cylindrical domain), and $\n$ is the unit outward normal vector on $\Gamma$. We assume throughout that $1 0$ is an arbitrary real number. It follows that $$(N - \eps(p-1) - p)\, \int_{\Omega} \frac{1}{(1+|x|)^{p}} |u|^{p}\, dx \leq \eps^{1-p} \, \int_{\Omega} |\nabla u |^{p}\, dx + \int_{\Gamma} \frac{|\n \cdot x|}{(1+|x|)^p} |u|^p \, d\Gamma \, ,$$ and for $\eps$ small enough, the desired inequality follows by standard density arguments. \bewende Now denote by $L^r(\Omega; w_1)$ and $L^q(\Gamma; w_2)$ the weighted Lebesgue spaces with weight functions $$w_i(x) = (1+|x|)^{\alpha_i}\, , \quad i=1,2, \quad \alpha_i \in \R \label{eq1a}$$ and norm defined by $$\| u \|_{r,w_1}^r = \int_{\Omega} w_1 |u(x)|^{r} \, dx \, , \qquad \mbox{and} \qquad \| u \|_{q,w_2}^q = \int_{\Gamma} w_2 |u(x)|^{q} \, dx \, .$$ Then we have the following embedding and trace theorem. \begin{theo} \label{t2} If $$p \leq r \leq \frac{pN}{N-p} \quad \mbox{and} \quad -N < \alpha_1 \leq r\, \frac{N-p}{p} - N \, , \label{eq2}$$ then the embedding $E \hookrightarrow L^r(\Omega; w_1)$ is continuous. If the upper bounds for $r$ in (\ref{eq2}) are strict, then the embedding is compact. If $$p \leq q \leq \frac{p(N-1)}{N-p} \quad \mbox{and} \quad -N < \alpha_2 \leq q\, \frac{N-p}{p} - N + 1 \, , \label{eq3}$$ then the trace operator $E \to L^q(\Gamma; w_2)$ is continuous. If the upper bounds for $q$ in (\ref{eq3}) are strict, then the trace is compact. \end{theo} This theorem is a consequence of Theorem 2 and Corollary 6 of \cite{7}. As a corollary of Lemma \ref{t1} and Theorem \ref{t2} we obtain \begin{lemma} \label{t3} Let $b$ satisfy $c/(1+|x|)^{p-1} \leq b(x) \leq C/(1+|x|)^{p-1}$ for some constants $0 < c \leq C$. Then $$\| u \|_b^p = \int_{\Omega} a(x) |\nabla u |^p\,dx + \int_{\Gamma} b(x) |u|^p\,d\Gamma$$ defines an equivalent norm on $E$. \end{lemma} \paragraph{Proof.} The inequality $\| u \|_E \leq C_1 \| u \|_b$ follows directly from Lemma \ref{t1}, while from Theorem \ref{t2} (setting $p=q$ and $\alpha_2 = -(p-1)$) we obtain \begin{eqnarray*} \| u \|_b^p & \leq & \| a \|_{L^{\infty}} \int_{\Omega} | \nabla u |^p dx + C \int_{\Gamma} |u|^p (1+ |x|)^{-(p-1)} d\Gamma \\ & \leq & \| a \|_{L^{\infty}} \int_{\Omega} | \nabla u |^p dx + C_2 \| u \|_E^p , \end{eqnarray*} which shows the desired equivalence. \bewende \paragraph{ Remark.} In special geometries the lower bound for $b$ required in Lemma \ref{t3} can be improved. In view of Lemma \ref{t1} it is sufficient to assume $b(x) \geq |\n \cdot x| / (1+|x|)^{p}$, where $\n \cdot x = |\n | | x| \cos \gamma$ and $\gamma$ is the angle between $x$ and $\n$. For a cylindrical domain $\Omega = B \times \R$, where $B \subset \R^{N-1}$ is bounded, we obtain $| \cos \gamma | \leq C_B / |x|$, with a constant $C_B$ depending only on the diameter of $B$. This shows that in cylindrical domains, Lemma \ref{t3} holds under the weaker assumption $$\frac{c}{(1+|x|)^{p}} \leq b(x) \leq \frac{C}{(1+|x|)^{p-1}} \, .$$ We shall assume throughout the paper that $b$ satisfies the assumption of Lemma~\ref{t3} so that we can use $\| \cdot \|_b$ as an equivalent norm in $E$. \section{The superlinear case} We make the following assumptions \begin{description} \item{A1 } $f$ and $g$ are Carath\'{e}odory functions on $\Omega \times \R$ and $\Gamma \times \R$, respectively, $f(\cdot ,0) = g(\cdot ,0) = 0 \,$ and \begin{eqnarray*} |f(x,s)| \leq f_{0}(x) + f_{1} (x)|s|^{r-1} & , & \; p\leq r p$such that$ \mu F(x,s) \leq f(x,s) s $,$\mu G(x,s) \leq g(x,s) s$for a.\ e.$ x \in \Omega $, resp.$x \in \Gamma$and every$s \in \R$. \item{A4 } One of the following conditions holds: \begin{description} \item{ a) } There is a nonempty open set$O \subset \Omega$with$F(x,s) > 0$for$(x,s) \in O \times (0,\infty) $\item{ b) } There is a nonempty open set$U \subset \Gamma$with$G(x,s) > 0$for$(x,s) \in U \times (0,\infty) $and$G$satisfies$\bar{\mu} G(x,s) \leq g(x,s) s$with some$\bar{\mu} >r$. \item{ c) }$G(x,s) > 0$for$(x,s) \in U \times (0,\infty) $and and there exist an open, nonempty subset$V \subset \Omega $,$\overline{V} \cap U \neq \emptyset$and a constant$C_F$, such that$F(x,u) \geq -C_F $on$ V \times (0,\infty)$. \end{description} \end{description} We denote by$N_f$,$N_F$,$N_{g}$,$N_{G}$the corresponding Nemytskii operators. Under the assumptions above we have the following result. \begin{lemma} \label{t5} The operators \begin{eqnarray*} N_f : L^{r}(\Omega; w_1) \to L^{r/(r-1)}(\Omega; w_1^{1/(1-r)}) \, , && N_F : L^{r}(\Omega; w_1) \to L^{1}(\Omega) \, , \\ N_{g} : L^{q}(\Gamma; w_2) \to L^{q/(q-1)}(\Gamma; w_2^{1/(1-q)}) \, , && N_{G} : L^{q}(\Gamma; w_2) \to L^{1}(\Gamma) \end{eqnarray*} are bounded and continuous. \end{lemma} \paragraph{Proof.} We only prove the statements for$N_g$and$N_G$, since the arguments for$N_f$and$N_F$are similar. Let$q^{\prime} = q/(q-1)$and$u \in L^{q}(\Gamma; w_2)$. Then, by Assumption A1, \begin{eqnarray*} \int_{\Gamma} | N_g (u) |^{q^{\prime}} w_2^{1/(1-q)} d\Gamma & \leq & 2^{q^{\prime}-1} \left( \int_{\Gamma} g_0^{q^{\prime}} w_2^{1/(1-q)} d\Gamma + \int_{\Gamma} g_1^{q^{\prime}} |u|^q w_2^{1/(1-q)} d\Gamma \right) \\ & \leq & 2^{q^{\prime}-1} \left( C + C_g \, \int_{\Gamma} |u|^q w_2 d\Gamma \right) , \end{eqnarray*} which shows that$N_g$is bounded. In a similar way we obtain \begin{eqnarray*} \int_{\Gamma} | N_G (u) | d\Gamma & \leq & \int_{\Gamma} g_0 | u | d\Gamma + \int_{\Gamma} g_1 |u|^q d\Gamma \\ & \leq & \left( \int_{\Gamma} g_0^{q^{\prime}} w_2^{1/(1-q)} d\Gamma \right)^{\frac{1}{q^{\prime}}} \left( \int_{\Gamma} | u |^q w_2 \, d\Gamma \right)^{\frac{1}{q}} + C_g \, \int_{\Gamma} | u |^q w_2 \, d\Gamma \end{eqnarray*} and again we claim that$N_G$is bounded. The continuity of these operators now follows from the usual properties of Nemytskii operators (cf. \cite{9}). \bewende \begin{lemma} \label{t6} Under Assumptions A1--A4,$J$is Fr\'{e}chet--differentiable on$E$and satisfies the Palais--Smale condition. \end{lemma} \paragraph{Proof.} We use the notation$I(u) = \frac{1}{p} \| u \|_b^p $,$K_F (u) = \int_{\Omega} F(x,u)\,dx $,$K_G (u) = \int_{\Gamma} G(x,u)\,d\Gamma $. Then the directional derivative of$J$in direction$h \in E$is $$\langle J^{\prime} u, h \rangle = \langle I^{\prime} u, h \rangle - \langle K_F^{\prime} u, h \rangle - \langle K_G^{\prime} u, h \rangle \, ,$$ where \begin{eqnarray*} &\langle I^{\prime}(u), h \rangle = \int_{\Omega} a(x) |\nabla u|^{p-2} \nabla u \nabla h \, dx + \int_{\Gamma} b(x) |u|^{p-2} u h \, d\Gamma \, , & \\ &\langle K_F^{\prime} (u), h \rangle = \int_{\Omega} f(x,u) h \, dx \, , \quad \langle K_G^{\prime} (u), h \rangle \; = \; \int_{\Gamma} g(x,u) h \, d\Gamma \, . & \end{eqnarray*} Clearly,$I^{\prime} : E \to E^{\prime}$is continuous. The operator$K_G^{\prime}$is a composition of operators $$K_G^{\prime} : E \to L^q(\Gamma; w_2) \stackrel{N_g}{\longrightarrow} L^{q/(q-1)}(\Gamma; w_2^{1/(1-q)}) \stackrel{\ell}{\longrightarrow} E^{\prime} ,$$ where$\langle \ell (v), h \rangle = \int_{\Gamma} v h \, d\Gamma $. Since $$\int_{\Gamma} |v h| \, d\Gamma \leq \left( \int_{\Gamma} |v|^{q^{\prime}} w_2^{1/(1-q)} d\Gamma \right)^{1/q^{\prime}} \left( \int_{\Gamma} |h|^q w_2 \, d\Gamma \right)^{1/q} ,$$$\ell$is continuous by Theorem \ref{t2}. As a composition of continuous operators,$K_G^{\prime}$is continuous, too. Moreover, by our assumptions on$w_2$(see A1), the trace operator$E \to L^q(\Gamma; w_2) \, $is compact and therefore,$K_G^{\prime}$is also compact. In a similar way we obtain that$ K_F^{\prime}$is compact and the Fr\'{e}chet-differentiability of$J$follows. Now let$u_k \in E$be a Palais--Smale sequence, i.\ e.$| J(u_k) | \leq C$for all$k$and$J^{\prime}(u_k) \to 0 $as$k \to \infty$. For$k$large enough we have$| \langle J^{\prime}(u_k), u_k \rangle | \leq \| u_k \|_b$and by Assumption A3 \begin{eqnarray*} C + \| u_k \|_b & \geq & J(u_k) - \frac{1}{\mu} \langle J^{\prime}(u_k), u_k \rangle \\ & \geq & \left( \frac{1}{p} - \frac{1}{\mu} \right) \, \| u \|_b^p \, . \end{eqnarray*} This shows that$u_k$is bounded in$E$. To show that$u_k$contains a Cauchy sequence we use the following inequalities for$\xi, \zeta \in \R^N$(see \cite{3}, Lemma 4.10): \begin{eqnarray} &| \xi - \zeta |^p \leq C ( |\xi |^{p-2} \xi - |\zeta |^{p-2}\zeta)(\xi - \zeta) \, , \quad \mbox{for } p \geq 2, &\label{eq21} \\ &| \xi - \zeta |^2 \leq C ( |\xi |^{p-2} \xi - |\zeta |^{p-2}\zeta)(\xi - \zeta) ( |\xi | + |\zeta | )^{2-p} \, , \quad \mbox{for } 1< p < 2\,.& \label{eq22} \end{eqnarray} Then we obtain in the case$p \geq 2$: \begin{eqnarray*} \| u_n - u_k \|^p_b & = & \int_{\Omega} a(x) |\nabla u_n - \nabla u_k |^{p} dx + \int_{\Gamma} b(x) |u_n - u_p |^{p} d\Gamma \\ & \leq & C \Big( \langle I^{\prime}(u_n), u_n - u_k \rangle - \langle I^{\prime}(u_k), u_n - u_k \rangle \Big) \\ & = & C \Big( \langle J^{\prime}(u_n), u_n - u_k \rangle - \langle J^{\prime}(u_k), u_n - u_k \rangle + \langle K_F^{\prime}(u_n) \\ &&+ K_G^{\prime}(u_n), u_n - u_k \rangle - \langle K_F^{\prime}(u_n) + K_G^{\prime}(u_k), u_n - u_k \rangle \Big) \\ & \leq & C \Big( \| J^{\prime}(u_n) \|_{E^{\prime}} + \| J^{\prime}(u_k) \|_{E^{\prime}} + \| K_F^{\prime}(u_n) - K_F^{\prime}(u_k) \|_{E^{\prime}} \\ & & + \| K_G^{\prime}(u_n) - K_G^{\prime}(u_k) \|_{E^{\prime}} \Big) \| u_n - u_k \|_b \,. \end{eqnarray*} Since$ J^{\prime}(u_k) \to 0$and$ K_F^{\prime}, K_G^{\prime}$are compact, there exists a subsequence of$u_k$which converges in$E$. If$1 < p < 2$, then we use (\ref{eq22}) and H\"older's inequality to obtain the estimate $$\| u_n - u_k \|^2_b \leq C \Big| \langle I^{\prime}(u_n), u_n - u_k \rangle - \langle I^{\prime}(u_k), u_n - u_k \rangle \Big| \Big( \| u_n \|_b^{2-p} + \| u_k \|_b^{2-p} \Big) .$$ Since$\| u_n \|_b$is bounded, the same arguments as above lead to a convergent subsequence. \bewende \begin{theo} \label{t7} There exists a nontrivial nonnegative solution of (\ref{p1}), (\ref{p2}) in$E$. \end{theo} \paragraph{Proof.} We shall use the Mountain--Pass lemma \cite{8} to obtain a solution. First we observe that, from Assumption A1 and A2, for every$\eps > 0$there is a$C_{\eps}$such that$| F(x, u) | \leq \eps f_0(x)| u|^p + C_{\eps}f_{1}(x)|u|^r $, and$| G(x,u) | \leq \eps g_0(x) |u|^p + C_{\eps} g_{1}(x)|u|^q $. Consequently \begin{eqnarray*} J(u) & \geq & \frac{1}{p} \| u \|^p_b - \int_{\Omega} \left( \eps f_0 (x) |u|^p + C_{\eps}f_{1}(x) |u|^r \right) dx \\ &&- \int_{\Gamma} \left( \eps g_0 (x) |u|^p + C_{\eps} g_{1}(x) |u|^q \right) d\Gamma \\ & \geq & \| u \|_b^p - \eps C_1 \| u \|_b^p - C_{\eps} C_2 (\| u \|_b^r + \| u \|_b^q ) \end{eqnarray*} and for$ \eps$and$\| u \|_b = \rho$sufficiently small, the right hand side is strictly greater than$0$. It remains to show that there exists$u_0 \in E, \; \| u_0 \|_b > \rho $such that$J(u_0) \leq 0$. In the case A4\ a), we choose a nontrivial nonnegative function$\vi \in C_0^{\infty}(O)$. From A3 we see that$F(x,s) \geq C_1 s^{\mu} - C_2$on$O \times (0,\infty)$. Then, for$t \geq 0$, $$J(t \vi) \leq \frac{1}{p} \, t^p \| \vi \|^p_b - C_1 t^{\mu} \int_{O} \vi^{\mu} dx + C_2 | O | \, .$$ Since$\mu > p$, the right hand side tends to$-\infty$as$t \to \infty$and for sufficiently large$t_0$,$u_0 = t_0 \vi$has the desired properties. In the case A4\ b), we choose a nonnegative$\vi \in C_{\delta}^{\infty}(\Omega)$such that$\mbox{supp} \vi \cap \Gamma \subset U$is not empty. Again from$G(x,s) \geq C_3 s^{\bar{\mu}} - C_4$on$U \times (0,\infty)$and Assumption A1 we claim $$J(t \vi) \leq \frac{1}{p} \, t^p \| \vi \|^p_b + C_5 \int_{\Omega} t \vi + t^r \vi^r dx - C_3 t^{\bar{\mu}} \int_{U} \vi^{\bar{\mu}} d\Gamma + C_4 | U | \, .$$ Since$\bar{\mu} > r \geq p$, we obtain$J(t \vi) \to -\infty$as$t \to \infty $. In the case A4\ c), we take$\vi \in C_{\delta}^{\infty}(\Omega)$with$\mbox{supp} \vi \cap \overline{\Omega} \subset \overline{V}$and$\mbox{supp} \vi \cap U \neq \emptyset $. Then $$J(t \vi) \leq \frac{1}{p} \, t^p \| \vi \|^p_b + C_F | V | - C_3 t^{\mu} \int_{U} \vi^{\mu} d\Gamma + C_4 | U |$$ and again we claim$J(t \vi) \to -\infty$as$t \to \infty $. Since$J$satisfies the Palais--Smale condition and$J(0) = 0$, the Mountain--Pass Lemma shows that there is a nontrivial critical point of$J$in$E$with critical value $$c = \inf_{\gamma \in P} \max_{t\in [0,1]} J(\gamma(t)) > 0 \, ,$$ where$P = \{ \gamma \in C([0,1], E) \mid \gamma(0) = 0, \, \gamma(1) = u_0 \}$. To obtain a nonnegative solution by this procedure, we introduce the truncated functions$\bar{f}$and$\bar{g}$such that$\bar{f}(x,s) = \bar{g}(x,s) = 0 $for all$s\leq 0$. Then the arguments above remain true and we obtain a critical point$u$of the truncated functional$\bar{J}$, i.\ e.$\langle \bar{J}^{\prime}(u), h \rangle = 0$for all$h \in E$. In particular, setting$u_{-}(x) = \max \{ -u(x), 0 \}$and$h = u_{-}$, we claim that$u \geq 0$. Since any nonnegative solution of the truncated problem is also a solution of the original equation, we have found a nonnegative solution of (\ref{p1}), (\ref{p2}). \bewende \section{Combined Sub- and Superlinear Nonlinearities} In this part we introduce an additional parameter into equation (\ref{p1}), i.\ e.\ we study $$-\diver (a(x) |\nabla u|^{p-2} \nabla u) = \lambda f(x,u) \quad \mbox{ in } \quad \Omega \eqno (1)_{\lambda}$$ with the same boundary condition (\ref{p2}) as before. Here, we assume the following \begin{description} \item{B1 } Let$g$satisfy Assumptions A1--A3 with$g_0 \equiv 0$and$|f(x,s)| \leq f_{1} (x)|s|^{r-1}, \quad 1 \leq r < p $, where$f_1$is nonnegative, measurable and there exists$\alpha_1$,$-N < \alpha_1 < r \frac{N-p}{p} - N $, such that for$w_1 (x) = (1 + |x|)^{\alpha_1} $, we have$f_1 \in L^{p/(p-r)}(\Omega; w_1^{r/(r-p)}) $. \item{B2 }$ |f(x,s)| \geq f_{2} (x)|s|^{\bar{r}-1}, \quad 1 \leq \bar{r} \leq r $, with$f_2 > 0$in some nonempty open set$O \subset \Omega$. \item{B3 } There is a nonempty open set$U \subset \Gamma$with$G(x,s) > 0$for$(x,s) \in U \times (0,\infty) $. \end{description} The Nemytskii operators$N_g$and$N_G$have the same properties as in Lemma \ref{t5}, while for$N_f$and$N_F$we obtain \begin{lemma} \label{t8} The operators$N_f : L^{p}(\Omega; w_1) \to L^{p/(p-1)}(\Omega; w_1^{1/(1-p)})$, and$N_F : L^{p}(\Omega; w_1) \to L^{1}(\Omega)$are bounded and continuous. \end{lemma} \paragraph{Proof.} Since the first statement is trivial if$r=1$, we may assume that$r>1$. From B1 we obtain with H\"older's inequality (setting$p^{\prime} = p/(p-1)$) \begin{eqnarray*} \int_{\Omega} | f(x,u) |^{p^{\prime}} w_1^{1/(1-p)} dx & \leq & \int_{\Omega} |f_1|^{p^{\prime}} w_1^{r/(1-p)} |u |^{p^{\prime}(r-1)} w_1^{(r-1)/(p-1)} dx \\ & \leq & \left( \int_{\Omega} |f_1|^{p/(p-r)} w_1^{r/(r-p)} \right)^{\frac{p-r}{p-1}} \left( \int_{\Omega} |u|^{p} w_1 \right)^{\frac{r-1}{p-1}} \\ & \leq & C \, \| u \|_{p,w_1}^{p(r-1)/(p-1)} \, . \end{eqnarray*} For$N_F$we obtain \begin{eqnarray*} \int_{\Omega} | F(x,u) | dx & \leq & \int_{\Omega} | f_1 | w_1^{-r/p} |u |^r w_1^{r/p} dx \\ & \leq & \left( \int_{\Omega} | f_1 |^{p/(p-r)} w_1^{r/(r-p)} dx \right)^{(p-r)/p} \left( \int_{\Omega} | u |^{p} w_1 dx \right)^{r/p} \\ & \leq & C \, \| u \|_{p,w_1}^{r} \, . \end{eqnarray*} \vspace*{-6mm} \bewende \vspace*{3mm} \noindent The differentiability for$J$now follows as above. To obtain the Palais--Smale condition for$J$, let$u_k \in E$be a sequence such that$ | J(u_k) | \leq C$and$J^{\prime}(u_k) \to 0 $as$k \to \infty$. With Assumptions A3, B1 and H\"older's inequality we get \begin{eqnarray*} \lefteqn{ J(u_k) - \frac{1}{\mu} \langle J^{\prime}(u_k), u_k \rangle } &&\\ & = & \left( \frac{1}{p} - \frac{1}{\mu} \right) \| u_k \|_b^p + \int_{\Omega} \frac{1}{\mu} f(x,u)u - F(x,u) \, dx + \int_{\Gamma} \frac{1}{\mu} g(x,u)u - G(x,u) \, d\Gamma \\ & \geq & \left( \frac{1}{p} - \frac{1}{\mu} \right) \| u_k \|_b^p - \left( 1 + \frac{1}{\mu} \right) \int_{\Omega} f_1(x) |u_k|^r dx \\ & \geq & \left( \frac{1}{p} - \frac{1}{\mu} \right) \| u_k \|_b^p - \left( \int_{\Omega} f_1^{p/(p-r)} w_1^{r/(r-p)} dx \right)^{(p-r)/p} \left( \int_{\Omega} |u_k|^{p} dx \right)^{r/p} \\ & \geq & \left( \frac{1}{p} - \frac{1}{\mu} \right) \| u_k \|_b^p - C_1 \| f_1 \|_{*} \| u_k \|_b^r \, , \end{eqnarray*} where$\| f_1 \|_{*}$is the weighted norm of$f_1$in$L^{p/(p-r)}(\Omega; w_1^{r/(r-p)}) $. Since$r 0$, such that for every$0 < \lambda < \lambda^{*}$, there are at least two nontrivial nonnegative solutions of (\ref{p1})$_{\lambda}$, (\ref{p2}). \end{theo} \paragraph{Proof.} First we show that for$\lambda \in (0, \lambda^{*})$, we can find$\rho > 0$such that$J(u) \geq c > 0 \, $if$\| u \|_b = \rho $. We denote by$C_{\Omega}, C_{\Gamma}$the embedding and trace constants for the operators$E \hookrightarrow L^p(\Omega; w_1)$and$E \to L^q(\Gamma; w_2) $, respectively. We obtain \begin{eqnarray*} J_{\lambda}(u) & \geq & \frac{1}{p} \| u \|_b^p - \frac{\lambda}{r} \int_{\Omega} f_1(x) |u|^r \, dx - \frac{1}{q} \int_{\Gamma} g_1(x) |u|^q \, d\Gamma \\ & \geq & \frac{1}{p} \| u \|_b^p - \frac{\lambda}{r} \left( \int_{\Omega} f_1(x)^{p/(p-r)} w_1(x)^{r/(r-p)} dx \right)^{(p-r)/p} \left( \int_{\Omega} |u|^p w_1 dx \right)^{r/p} \\ &&- \frac{1}{q} \int_{\Gamma} g_1(x) |u|^q \, d\Gamma \\ & \geq & \frac{1}{p} \| u \|_b^p - \frac{\lambda}{r} C_{\Omega} \| f_1 \|_{*} \| u \|_b^r - \frac{1}{q} C_{\Gamma} C_{g} \| u \|_b^{q} \, . \end{eqnarray*} If$\| u \|_b = \rho$, we obtain $$\label{eq32} J_{\lambda}(u) \geq \frac{1}{p}\, \rho^p \left( 1 - \frac{p \lambda}{r} C_{\Omega} \| f_1 \|_{*} \rho^{r - p} - \frac{p}{q}\, C_{\Gamma} C_{g} \rho^{q-p} \right)$$ Elementary calculations show that the right hand side is maximal for $$\rho_m = \left( \frac{q(p-r) \lambda \, C_{\Omega} \| f_1 \|_{*} }{r(q-p) \, C_g C_{\Gamma}} \right)^{1/(q-r)} \, .$$ Inserting this into equation (\ref{eq32}), we find that the right hand side is zero for $$\lambda = \lambda^{*} := \left[ \frac{p}{r}\, \| f_1 \|_{*} C_{\Omega} C_0^{\frac{r-p}{q-r}} + \frac{p}{q}\, C_g C_{\Gamma} C_0^{\frac{q-p}{q-r}} \right]^{\frac{r-q}{q-p}} \, ,$$ where $$C_0 = \left( \frac{\| f_1 \|_{*} C_{\Omega} (p-r)q}{C_g C_{\Gamma}(q-p)r} \right) \, ,$$ and strictly greater than 0 for$\lambda < \lambda^{*}$. This shows that for every$\lambda < \lambda^{*}$, we find$\rho_{\lambda} > 0$such that$J_{\lambda} \geq c_{\lambda} > 0$for$\| u \|_b = \rho_{\lambda}$. The existence of a function$u_0 \in E, \, \| u_0 \|_b > \rho_{\lambda}$and$J_{\lambda}(u_0) \leq 0$now follows as in the proof of Theorem 2 (case A4\ b). Then the Mountain-Pass Lemma again implies the existence of a nontrivial solution$u_1$with$J_{\lambda}(u_1) \geq c_{\lambda}$. On the other hand, for$\vi \in C_0^{\infty}(O)$and$t > 0$we obtain $$J_{\lambda}(t \vi) \leq \frac{t^p}{p} \| \vi \|_b^p - \frac{t^{\bar{r}}}{\bar{r}}\, \int_{O} f_2(x) | \vi |^{\bar{r}} dx \, .$$ This shows that$J_{\lambda}(t \vi) < 0$for sufficiently small$t$and consequently$J_{\lambda}$attains its minimum in the ball$B_{\rho_{\lambda}} \subset E$. We claim that there is a second solution$u_2 \in B_{\rho_{\lambda}}$with$J_{\lambda}(u_2) < 0$. In addition, with the same truncation procedure as in the proof of Theorem 2, we claim that there are two nonnegative solutions. \bewende Now we can prove the corresponding result for equation (\ref{p1}) with boundary condition $$\n \cdot a(x) |\nabla u|^{p-2} \nabla u + b(x) |u|^{p-2}u = \lambda g(x,u) \quad \mbox{ on } \quad \Gamma \eqno (2)_{\lambda}$$ if we interchange the roles of$g$and$f$in Assumptions B1--B3. That is, we assume now that$f$satisfies Assumptions A1--A4\ a) (with$f_0 \equiv 0$) and$g$satisfies \begin{description} \item{B4 }$| g(x,s)| \leq g_1(x) |s|^{q-1}$,$1 \leq q < p$,$g_1 \in L^{p/(p-q)}(\Gamma; w_2^{q/(q-p)})$,$| g(x,s)| \geq g_2(x) |s|^{\bar{q}-1}$,$1 \leq \bar{q} \leq q$and$g_2 > 0$in some nonempty open set$ U \subset \Gamma $. \end{description} \begin{theo} \label{t10} Let$f$satisfy Assumptions A1--A4\ a) (with$f_0 \equiv 0$) and$g$satisfy B4. Then for every$0 < \lambda < \lambda^{*}$, there are at least two nontrivial nonnegative solutions of (\ref{p1}), (\ref{p2})$_{\lambda}$. \end{theo} \paragraph{Proof.} First we claim as in Lemma \ref{t8} that $$N_g : L^{p}(\Gamma; w_2) \to L^{p/(p-1)}(\Gamma; w_2^{1/(1-p)}) \; , \qquad N_G : L^{p}(\Gamma; w_2) \to L^{1}(\Gamma)$$ are bounded and continuous. The estimate for$J_{\lambda}$now reads $$J_{\lambda}(u) \geq \frac{1}{p} \| u \|_b^p - \frac{1}{r} C_{\Omega} C_f \| u \|_b^{r} - \frac{\lambda}{q} C_{\Gamma} \| g_1 \|_{*} \| u \|_b^{q} ,$$ where$\| g_1 \|_{*}$is the norm of$g_1$in$L^{p/(p-q)}(\Gamma; w_2^{q/(q-p)}) $. Now$\lambda^{*}$can be calculated as $$\lambda^{*} := \left[ \frac{p}{q}\, \| g_1 \|_{*} C_{\Gamma} \bar{C}_0^{\frac{q-p}{r-q}} + \frac{p}{r}\, C_f C_{\Omega} \bar{C}_0^{\frac{r-p}{r-q}} \right]^{\frac{q-r}{r-p}} \, , \quad \bar{C}_0 = \left( \frac{\| g_1 \|_{*} C_{\Gamma} (p-q)r}{C_f C_{\Omega}(r-p)q} \right) \, .$$ The existence of$u_0$with$\| u_0 \|_b > \rho_{\lambda}$and$J(u_0) < 0 $follows in the same way as in the proof of Theorem \ref{t7}, case A4\ a). Finally, for a nonnegative$\vi \in C_{\delta}^{\infty}(\Omega)$with$\mbox{supp}\,\vi \cap \Gamma \subset U$not empty, we find $$J_{\lambda}(t \vi) \leq \frac{t^p}{p} \| \vi \|_b^p + C \frac{t^r}{r} \| \vi \|_b^r - \frac{t^{\bar{q}}}{\bar{q}} \int_{U} g_2(x) | \vi |^{\bar{q}} dx \, .$$ Since$ \bar{q} < p \leq r $,$J_{\lambda}(t \vi) < 0$for$t$sufficiently small and we claim that$J_{\lambda}$attains its minimum in$ B_{\rho_{\lambda}} \subset E$. \bewende We remark that, if$\Omega$is of class$C^{1,\alpha}\, (\alpha \leq 1)$and, in addition to B4,$g$satisfies $$| g(x,s) - g(y, t) | \leq C \Big( |x-y|^{\alpha} + |s-t|^{\alpha} \Big) , \qquad |g(x,s)| \leq C$$ for all$x,y \in \Gamma$,$s,t \in \R$, then the regularity result of \cite{4}, Thm.\ 2, shows that the solution$u$belongs to$C^{1,\beta}(\overline{\Omega})$for some$\beta > 0$. \begin{thebibliography}{99} \bibitem{1} A.\ Ambrosetti, H.\ Brezis and G.\ Cerami, {\em Combined effects of concave and convex nonlinearities in some elliptic problems.} J. 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