Laplace's equation with concave and convex boundary nonlinearities on an exterior region

Abstract

This paper studies Laplace's equation $-\Delta u=0$ in an exterior region $U{\mathbb{R}}^N$, when $N\ge 3$, subject to the nonlinear boundary condition $\frac{\partial u}{\partial \nu }=\lambda {\left|u\right|}^{q-2}u+\mu {\left|u\right|}^{p-2}u$ on $\partial U$ with $1<q<2<p<2^{\ast }$. In the function space $\mathcal{H}\left(U\right)$, one observes when $\lambda >0$ and $\mu \in \mathbb{R}$ arbitrary, then there exists a sequence $\left\{u_k\right\}$ of solutions with negative energy converging to $0$ as $k\to \infty$; on the other hand, when $\lambda \in \mathbb{R}$ and $\mu >0$ arbitrary, then there exists a sequence $\left\{{\tilde{u}}_k\right\}$ of solutions with positive and unbounded energy. Also, associated with the $p$-Laplacian equation $-{\Delta }_{p}u=0$, the exterior $p$-harmonic Steklov eigenvalue problems are described.