A nonlinear elliptic equation with singular potential and applications to nonlinear field equations

Abstract

We prove the existence of cylindrical solutions to the semilinear elliptic problem -Δu + u/|y|2 = f(u), u ∈ H1(ℝ N), u ≥ 0, where (y, z) ∈ ℝk × ℝN-k, N > k ≥ 2 and f has a double-power behaviour, subcritical at infinity and supercritical near the origin. This result also implies the existence of solitary waves with nonvanishing angular momentum for nonlinear Schrödinger and Klein-Gordon equations. © European Mathematical Society 2007.