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We deal with the following semilinear equation in exterior domains \[-\Deltau + u = a(x)|u|^{p-2}u,\qquad u\in H^1_0({A_R}), \] 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 exhibita positive solution having the same features as $a$, for values of $p>2$ in asuitable range that includes exponents greater than the standard Sobolevcritical one. In the special case of radial weight $a$, our existence resultensures multiplicity of nonradial solutions. We also provide an existenceresult for supercritical $p$ in nonradial exterior domains.

Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains