Multiplicity of Solutions for Fourth-Order Elliptic Equations with p-Laplacian and Mixed Nonlinearity

Abstract

In the paper, we study the multiplicity of solutions for a class of fourth-order elliptic equations with p-Laplacian and mixed nonlinearity of the form:

$$\begin{aligned} \left\{ \begin{array}{lll} \Delta ^2u-\Delta _{p}u+\lambda V(x)u=f(x,u)+\mu \xi (x)\vert u\vert ^{q-2}u, \ \ x\in {\mathbb {R}}^N, \\ u\in H^{2}({\mathbb {R}}^N). \\ \end{array} \right. \end{aligned}$$

Unlike most other works, we replace the Laplacian with a p-Laplacian. Using the mountain pass theorem and Ekeland’s variational principle, we establish the existence of two nontrivial solutions. To overcome the difficulty of the convergence of the subsequences for the Palais–Smale sequences of the Euler–Lagrange functional, we consider Cerami sequences. Our results extend the recent results of Zhang et al. (Electron J Differ Equ 2017(250):1–15, 2017).