Multiplicity for nonlinear elliptic boundary value problems of p-Laplacian type without Ambrosetti-Rabinowitz condition

Abstract

In this paper, we study the existence of multiple solutions to the following nonlinear elliptic boundary value problem of p-Laplacian type: $$\left\{ \begin{gathered} - \Delta _p u = f(x,u),x \in \Omega , \hfill \\ u = 0,x \in \partial \Omega , \hfill \\ \end{gathered} \right. $$ (*) where 1 < p < ∞, Ω ⊆ ℝN is a bounded smooth domain, Δpu = div(|Du|p−2Du) is the p-Laplacian of u and f: Ω × ℝ → ℝ satisfies \(\mathop {\lim }\limits_{|t| \to \infty } \tfrac{{f(x,t)}} {{|t|^{p - 2} t}} = l \) uniformly with respect to x ∈ Ω, and l is not an eigenvalue of −Δp in W01,p(Ω) but f(x, t) dose not satisfy the Ambrosetti-Rabinowitz condition. Under suitable assumptions on f(x, t), we have proved that (*) has at least four nontrivitial solutions in W01,p(Ω) by using Nonsmooth Mountain-Pass Theorem under (C)c condition. Our main result generalizes a result by N. S. Papageorgiou, E. M. Rocha and V. Staicu in 2008 (Calculus of Variations and Partial Differential Equations, 33: 199–230(2008)) and a result by G. B. Li and H. S. Zhou in 2002 (Journal of the London Mathematical Society, 65: 123–138(2002)).