Nonlinear elliptic equations with subhomogeneous potentials

Abstract

We prove the existence of nonnegative symmetric solutions to the semilinear elliptic equation − △ u + V ( | y 1 | , … , | y k | ) u = g ( u ) in R N where x = ( z , y 1 , … , y k ) ∈ R N 0 × R N 1 × ⋯ × R N k = R N with N ≥ 3 , k ≥ 1 , N 0 ≥ 0 and N i ≥ 2 for i > 0 . The nonlinearity g and the potential V are, respectively, a continuous function, not necessarily superlinear at infinity, and a positive measurable function, not necessarily homogeneous but satisfying a subhomogeneity condition, which implies vanishing at infinity and singularity at least at the origin. This also yields the existence of nonrotating solitary waves and vortices with a critical frequency for nonlinear Schrödinger and Klein–Gordon equations with singular cylindrical potentials.