Multi-bump solutions for the nonlinear magnetic Schr\"odinger equation with logarithmic nonlinearity

Abstract

In this paper, we study the following nonlinear magnetic Schr\"odinger equation with logarithmic nonlinearity \begin{equation*} -(\nabla+iA(x))^2u+\lambda V(x)u =|u|^{q-2}u+u\log |u|^2,\ u\in H^1(\mathbb{R}^N,\mathbb{C}), \end{equation*} where the magnetic potential $A \in L_{l o c}^2\left(\mathbb{R}^N, \mathbb{R}^N\right)$, $2<q<2^*,\ \lambda>0$ is a parameter and the nonnegative continuous function $V: \mathbb{R}^N \rightarrow \mathbb{R}$ has the deepening potential well. Using the variational methods, we obtain that the equation has at least $2^k-1$ multi-bump solutions when $\lambda>0$ is large enough.