Standing waves for supercritical nonlinear Schrödinger equations

Let V(x) be a non-negative, bounded potential in , N 3 and p supercritical, . We look for positive solutions of the standing-wave nonlinear Schrödinger equation , with u(x)→0 as x→+∞. We prove that if V(x)=o(x−2) as x→+∞, then for N 4 and this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(x−μ) with μ>N, then this result still holds provided that N 3 and . Other conditions for solvability, involving behavior of V at ∞, are also provided.