Multiplicity and concentration of solutions for fourth-order elliptic equations with mixed nonlinearity

Abstract

This article concerns the fourth-order elliptic equation $$\displaylines{ \Delta^2u-\Delta u+\lambda V(x)u=f(x, u)+\mu \xi(x)|u|^{p-2}u, \quad x\in \mathbb{R}^{N},\cr u\in H^2(\mathbb{R}^{N}), }$$ where $\lambda >0$ is a parameter, $V\in C(\mathbb{R}^{N},\mathbb{R})$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of two nontrivial solutions. Moreover, the concentration of these solutions is explored on the set $V^{-1}(0)$ as $\lambda\to\infty$. As an application, we give the similar results and concentration phenomenona for the above problem with concave and convex nonlinearities.

Keywords

• Fourth-order elliptic equations
• concentration
• mixed nonlinearity
• concave-convex nonlinearity
• variational methods