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

TL;DR

This paper investigates solutions to Laplace's equation with nonlinear boundary conditions in exterior regions, revealing existence of solution sequences with specific energy behaviors and exploring related p-harmonic Steklov eigenvalue problems.

Contribution

It introduces new existence results for solutions with varying energy levels under nonlinear boundary conditions and studies associated p-harmonic Steklov eigenvalue problems.

Findings

Existence of solution sequences with negative energy approaching zero.

Existence of unbounded positive energy solutions.

Analysis of p-harmonic Steklov eigenvalue problems.

Abstract

This paper studies Laplace's equation $-\Delta\,u=0$ in an exterior region $U\varsubsetneq{\mathbb R}^N$, when $N\geq3$, subject to the nonlinear boundary condition $\frac{\partial u}{\partial\nu}=\lambda{\left\vert{u}\right\vert}^{q-2}u+\mu{\left\vert{u}\right\vert}^{p-2}u$ on $\partial U$ with $1<q<2<p<2^*$. In the function space $\mathscr{H}\left(U\right)$, 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.