Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains

TL;DR

This paper investigates the existence and multiplicity of solutions to supercritical elliptic equations in exterior domains, highlighting symmetry-breaking phenomena and extending results to nonradial and supercritical cases.

Contribution

It introduces new existence results for supercritical elliptic problems in exterior domains, including nonradial solutions and symmetry-breaking effects.

Findings

Existence of positive solutions with symmetry properties.

Multiplicity of nonradial solutions in radial weights.

Extension to supercritical exponents in nonradial domains.

Abstract

We deal with the following semilinear equation in exterior domains \[-\Delta u + u = a(x)|u|^{p-2}u,\qquad u\in H^1_0({A_R}), \] where ${A_R} := \{x\in\mathbb{R}^N:\, |x|>{R}\}$, $N\ge 3$, $R>0$. Assuming that the weight $a$ is positive and satisfies some symmetry and monotonicity properties, we exhibit a positive solution having the same features as $a$, for values of $p>2$ in a suitable range that includes exponents greater than the standard Sobolev critical one. In the special case of radial weight $a$, our existence result ensures multiplicity of nonradial solutions. We also provide an existence result for supercritical $p$ in nonradial exterior domains.