Multiple positive solutions for the $N$-Laplace equation with nonlinear Neumann boundary conditions

Abstract

Let $ {\Omega}\subset {\mathbb R}^N$ be a bounded domain with $C^2$ boundary. Let $u\in W^{1,N}(\Omega)$ be a weak solution of the following problem: $$(P_{\mu,{\lambda}})\hspace{1cm} \left \{ \begin{array}{cllll} \left . \begin{array}{rllll} -\text{div}(|{\nabla} u|^{N-2}{\nabla} u) +|u|^{N-2}u & =& \mu h(u) e^{u^{\alpha}}\\ u&<& 0 \end{array} \right \} \;\; \text{in} \; {\Omega}, \\ \hspace{2.2cm}|{\nabla} u|^{N-2}\frac{{\partial} u}{{\partial} \nu}\; =\;\; {\lambda} \psi u^q \;\;\text{on}\;\; \partial {\Omega}, \end{array} \right . $$ where ${\alpha}\in (0,\frac{N}{N-1}], {\lambda}, \mu >0, q\in [0,N-1)$ and $\psi$ is a positive Hölder continuous function on $\overline{ {\Omega}}$. Here, $h(u)$ is a ``suitable" perturbation of $e^{u^\alpha}$ as $u \to \infty$ (see assumptions $(\mathbf{A1})-(\mathbf{A5})$ in Section 1). In this article, we show that there exists a region $\Re \subset \{(\mu,{\lambda}): \mu,{\lambda}>0\}$ bounded by the graph of a map $ {\Lambda}$ such that $(P_{\mu,{\lambda}})$ admits at least two solutions for all $(\mu,{\lambda}) \in \Re$, at least one solution for any $(\mu,{\lambda}) \in \partial \Re$ and no solution for $(\mu,{\lambda})$ outside $\overline{\Re}$.