Multi-bump solutions for the nonlinear magnetic Schrödinger equation with logarithmic nonlinearity

In this paper, we study the following nonlinear magnetic Schrödinger equation with logarithmic nonlinearity \begin{equation*} -(\nabla+iA(x))^2u+λV(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 $A\in L_{loc}^2({\mathbb{R}}^N,{\mathbb{R}}^N)$, $2<q<2^{\ast },\lambda >0$ is a parameter and the nonnegative continuous function $V:{\mathbb{R}}^N\to \mathbb{R}$ 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.