Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations

Abstract

In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on RN, when N≥2,(0.1)−ΔNu+V(x)|u|N−2u=λ|u|r−2u+f(x,u). Here, V(x)>0:RN→R is a suitable potential function, r∈(1,N), f(x,u) is a continuous function of N-superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition, while λ>0 is a constant. A suitable Moser–Trudinger inequality and the compact embedding WV1,N(RN)↪Lr(RN) are proved to study problem (0.1). Moreover, the compact embedding HV1(RN)↪LKt(RN) is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation(0.2)−Δu+V(x)u=K(x)g(u) with potentials vanishing at infinity in a measure-theoretic sense when N≥3.