The Journal of Geometric Analysis ( IF 1.2 ) Pub Date : 2021-06-17 , DOI: 10.1007/s12220-021-00668-3 Yiwen Ma , Chao Ji
This paper concerns the following magnetic Schrödinger–Poisson system
$$\begin{aligned} {\left\{ \begin{array}{ll} -(\nabla +i A(x))^{2}u+(\lambda V(x)+Z(x))u+\phi u=f(\left| u\right| ^{2})u ,&{} \text { in }{{\mathbb {R}}}^{3}, \\ -\varDelta \phi = u^{2}, &{} \text { in } {\mathbb {R}}^{3}, \end{array}\right. } \end{aligned}$$where \(\lambda \) is a positive parameter, f has subcritical growth, the potentials V, \(Z:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}\) are continuous functions verifying some conditions, the magnetic potential \(A \in L_\mathrm{loc}^{2}({\mathbb {R}}^{3}, {\mathbb {R}}^{3}) \). Assuming that the zero set of V(x) has several isolated connected components \(\varOmega _{1},\ldots ,\varOmega _{k}\) such that the interior of \(\varOmega _{j}\) is non-empty and \(\partial \varOmega _{j}\) is smooth, where \(j\in \left\{ 1,\ldots ,k\right\} \), then for \(\lambda >0\) large enough, we show that the above system has at least \(2^{k}-1\) multi-bump solutions by using variational methods.
中文翻译：
$$\pmb {{\mathbb {R}}}^{3}$$ R 3 中磁性薛定谔-泊松系统的多凸点解的存在性
本文涉及以下磁薛定谔-泊松系统
$$\begin{aligned} {\left\{ \begin{array}{ll} -(\nabla +i A(x))^{2}u+(\lambda V(x)+Z(x))u+ \phi u=f(\left| u\right| ^{2})u ,&{} \text { in }{{\mathbb {R}}}^{3}, \\ -\varDelta \phi = u^{2}, &{} \text { in } {\mathbb {R}}^{3}, \end{array}\right。} \end{对齐}$$其中\(\lambda \)是一个正参数，f有亚临界增长，势V , \(Z:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}\)是连续函数验证一些条件，磁势\(A \in L_\mathrm{loc}^{2}({\mathbb {R}}^{3}, {\mathbb {R}}^{3}) \)。假设V ( x )的零集有几个孤立的连通分量\(\varOmega _{1},\ldots ,\varOmega _{k}\)使得\(\varOmega _{j}\) 的内部是非空的，并且\(\partial \varOmega _{j}\)是平滑的，其中\(j\in \left\{ 1,\ldots ,k\right\} \)，那么对于\(\lambda >0\)足够大，我们通过使用变分方法证明上述系统至少有\(2^{k}-1\) 个多凸点解。
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