Existence of solution for a generalized quasilinear elliptic problem

Abstract

It establishes existence and multiplicity of solutions to the elliptic quasilinear Schrödinger equation −div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=h(x,u),x∈ℝN,where g, h, V are suitable smooth functions. The function g is asymptotically linear at infinity and, for each fixed x∈ℝN, the function h(x, s) behaves like s at the origin and s3 at infinity. In the proofs, we apply variational methods.