Multi-bump Solutions for the Quasilinear Choquard Equation in $\mathbb {R}^{N}$

Abstract

This paper deal with the following quasilinear Choquard equation in $\mathbb {R}^{N}$:

$ \left \{\begin {array}{l} -{\Delta } u-u{\Delta }u^{2} +(\lambda V(x)+1) u = \left (\frac {1}{|x|{\!}^{\mu }} *|u|{\!}^{p}\right )|u|{\!}^{p-2} u, \quad x \in \mathbb {R}^{N}, \\ u \in H^{1}\left (\mathbb {R}^{N}\right ), \end {array}\right . $

where $0<\mu <\min \limits \left \{ 2, 8-2N\right \} , 2\le N<4, p \in \left [4, \frac {2N-\mu }{N-2}\right ),$ and λ > 0 is a real parameter. Here, $2^{*}=\frac {2 N}{N-2}$ if $N \geq 3,2^{*}=+\infty$, if N = 2. The potential V: $\mathbb {R}^{N} \rightarrow \mathbb {R}$ is a nonnegative continuous function verifying some assumptions. Using variational methods, we show that if ${\Omega }:=\text {int}V^{-1} \left (0 \right )$ has several isolated connected components Ω₁,⋯ ,Ω_k satisfying the interior of Ω_j is non-empty and that ∂Ω_j is smooth, thus for λ > 0 large enough, the above equation has at least 2^k − 1 multi-bump solutions.