Nonlinear elliptic equations with subhomogeneous potentials

We prove the existence of nonnegative symmetric solutions to the semilinear elliptic equation −△u+V(|y1|,…,|yk|)u=g(u)in RN where x=(z,y1,…,yk)∈RN0×RN1×⋯×RNk=RNx=(z,y1,…,yk)∈RN0×RN1×⋯×RNk=RN with N≥3N≥3, k≥1k≥1, N0≥0N0≥0 and Ni≥2Ni≥2 for i>0i>0. The nonlinearity gg and the potential VV 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.