Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian

Abstract

We investigate a class of nonlinear biharmonic equations with p-Laplacian{Δ2u−βΔpu+λV(x)u=f(x,u)in RN,u∈H2(RN), where N≥1, β∈R, λ>0 is a parameter and Δpu=div(|∇u|p−2∇u) with p≥2. Unlike most other papers on this problem, we replace Laplacian with p-Laplacian and allow β to be negative. Under some suitable assumptions on V(x) and f(x,u), we obtain the existence and multiplicity of nontrivial solutions for λ large enough. The proof is based on variational methods as well as Gagliardo–Nirenberg inequality.