Existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger-Kirchhoff-type equations

In this paper, we are concerned with the following Schrödinger-Kirchhoff-type problem:(P){−(a+b∫RN|∇u|pdx)p−1Δpu+λV(x)|u|p−2u=f(x,u),x∈RN,u∈W1,p(RN), where a>0, b≥0 are constants, Δpu:=div(|∇u|p−2∇u) is the p-Laplacian operator with p≥2, V(x) is the potential function satisfying some conditions which may not guarantee the compactness of the corresponding Sobolev embedding. By using the variational methods, we prove the existence and multiplicity of nontrivial solutions for problem (P).