Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains

Abstract

We deal with the following semilinear equation in exterior domains \[-Δu + u = a(x)|u|^{p-2}u,\qquad u\in H^1_0({A_R}), \] 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.

In the new version, we fixed a few inaccuracies