Regular solutions to a supercritical elliptic problem in exterior domains

TL;DR

This paper proves the existence of infinitely many regular solutions to a supercritical elliptic problem in exterior domains for small parameter values, expanding understanding of nonlinear elliptic equations in unbounded regions.

Contribution

It establishes the existence of infinitely many solutions to a supercritical elliptic PDE in exterior domains for small bb, a result not previously known.

Findings

Existence of infinitely many solutions for small bb.

Solutions are regular and satisfy zero Dirichlet boundary conditions.

Results apply to dimensions N a7 3.

Abstract

We consider the supercritical elliptic problem -\Delta u = \lambda e^u, \lambda > 0, in an exterior domain $\Omega = \mathbb{R}^N \setminus D$ under zero Dirichlet condition, where D is smooth and bounded in \mathbb{R}^N, N greater or equal than 3. We prove that, for \lambda small, this problem admits infinitely many regular solutions.