Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains

We deal with the following semilinear equation in exterior domains − Δ u + u = a ( x ) | u | p − 2 u , u ∈ H 0 1 ( A R ) , where A R := { x ∈ R N : | x | > R } , N ⩾ 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.