Existence and Concentration of Solutions for a 1-Biharmonic Choquard Equation with Steep Potential Well in ${{\textbf{R}}}^{N}$

Abstract

In this paper, we investigate the existence and concentration of solutions for the following 1-biharmonic Choquard equation with steep potential well

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta _{1}^{2}-\Delta _{1} u+(1+\lambda V(x))\frac{u}{|u|}=\left( I_{\mu } * F(u)\right) f(u) &{} \text {in }\mathbb {R}^{N}, \\ u\in {\text {BL}}(\mathbb {R}^{N}), \end{array}\right. } \end{aligned}$$

where $N\ge 3$, $\lambda >0$ is a positive parameter, $V:\mathbb {R}^N\rightarrow \mathbb {R}$, $f:\mathbb {R}\rightarrow \mathbb {R}$ are continuous functions verifying further conditions, $\Omega ={\text {int}}(V^{-1}(\{0\}))$ has nonempty interior and $I_{\mu }:\mathbb {R}^{N}\rightarrow \mathbb {R}$ is the Riesz potential of order $\mu \in (N-1, N)$. For $\lambda >0$ large enough, we prove the existence of a nontrivial solution $u_{\lambda }$ of the problem above via variational methods and the concentration behavior of $u_{\lambda }$ which is explored on the set $\Omega$.