Multiple solutions for generalized biharmonic equations with singular potential and two parameters

Abstract

We investigate a more general nonlinear biharmonic equation Δ 2 u − β Δ p u + V λ ( x ) u = f ( x , u ) in ℝ N , where Δ2:=Δ(Δ) is the biharmonic operator, N≥1, λ>0 and β∈ℝ are parameters, Δpu= div(|∇u|p−2∇u) with p≥2. Differently from previous works on biharmonic problems, we replace Laplacian with p-Laplacian, and suppose that V(x)=λa(x)−b(x) with λ>0 and b(x) can be singular at the origin, in particular we allow β to be a real number. Under suitable conditions on Vλ(x) and f(x,u), the multiplicity of solutions is obtained for λ>0 sufficiently large. Our analysis is based on variational methods as well as the Gagliardo–Nirenberg inequality.