On multi-bump solutions for the Choquard-Kirchhoff equations in $ \mathbb{R}^{N} $
Shuaishuai Liang, Mingzhe Sun, Shaoyun Shi, Sihua Liang
Abstract
In this paper, we investigate the following Choquard-Kirchhoff equations in $ \mathbb{R}^{N} $:$ \begin{eqnarray*} \left\{\begin{array}{l} - K_b(u) + (\lambda V(x)+1)|u|^{p-2}u = \left(\frac{1}{|x|^{\mu}} *F(u)\right)f(u)\ \ \mbox{in } \,\mathbb{R}^{N},\\ u \in W^{1,p}(\mathbb{R}^{N}), \end{array}\right. \end{eqnarray*} $where $ K_b(u) = \biggl(1+b \int_{\mathbb{R}^{N}}|\nabla u|^{p} dx\biggl)\Delta_pu $, $ \Delta_p $ is the $ p $-Laplacian operator, $ 0 < \mu < p < N $, $ b, \lambda $ are some positive parameters, $ f: \mathbb{R} \rightarrow \mathbb{R} $ is a continuous function, and the potential $ V: \mathbb{R}^{N} \rightarrow \mathbb{R} $ is a nonnegative continuous function. Under some proper conditions, by using the variational methods, we prove that the existence of multi-bump solutions for this problem if $ \Omega: = \mbox{int}V^{-1} \left ( 0 \right) $ has several isolated connected components $ \Omega_{1}, \cdots, \Omega_{k} $ satisfying the interior of $ \Omega_{j} $ is non-empty and $ \partial \Omega_{j} $ is smooth. Moreover, as $ \lambda>0 $ large enough, the above equation has at least $ 2^{k}-1 $ multi-bump solutions.