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

Jun Wang and Zhaoyang Yin

Mathematische Nachrichten, 2025, vol. 298, issue 1, 328-355

Abstract: In this paper, we study the following nonlinear magnetic Schrödinger equation with logarithmic nonlinearity −(∇+iA(x))2u+λV(x)u=|u|q−2u+ulog|u|2,u∈H1(RN,C),$$\begin{equation*} \hspace*{24pt}-(\nabla +\text{i}A(x))^2u+\lambda V(x)u =|u|^{q-2}u+u\log |u|^2,\ u\in H^1(\mathbb {R}^N,\mathbb {C}), \end{equation*}$$where the magnetic potential A∈Lloc2RN,RN$A \in L_{l o c}^2\left(\mathbb {R}^N, \mathbb {R}^N\right)$, 2 0$2 0$ is a parameter and the nonnegative continuous function V:RN→R$V: \mathbb {R}^N \rightarrow \mathbb {R}$ has the deepening potential well. Using the variational methods, we obtain that the equation has at least 2k−1$2^k-1$ multi‐bump solutions when λ>0$\lambda >0$ is large enough.

Date: 2025
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