Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains

We deal with the following semilinear equation in exterior domains $$-\mathrm{\Delta }u+u=a(x)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,u\in H_0^1(A_R),$$ where $A_R:=\{x\in {\mathbb{R}}^N:\mathrm{\mid }x\mathrm{\mid }>R\}$, $N\ge 3$, $R>0$. 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.