  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.
