Abstract In this paper, we study the following nonlinear magnetic Schrödinger-Poisson type equation (εi∇−A(x))2u+V(x)u+ϵ−2(|x|−1∗|u|2)u=f(|u|2)uin R3,u∈H1(R3,C), $$\begin{array}{} \displaystyle \left\{\!\begin{aligned}&\Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u = f(|u|^{2})u\quad\hbox{in }\mathbb{R}^3,\\&u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}),\end{aligned}\right. \end{array}$$ where ϵ > 0, V : ℝ3 → ℝ and A : ℝ3 → ℝ3 are continuous potentials. Under a local assumption on the potential V, by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration properties of nontrivial solutions for ε > 0 small. In this problem, the function f is only continuous, which allow to consider larger classes of nonlinearities in the reaction.

中文翻译：

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一类磁性薛定谔-泊松方程的集中解的多重性

摘要 在本文中，我们研究了以下非线性磁薛定谔-泊松型方程 (εi∇−A(x))2u+V(x)u+ϵ−2(|x|−1∗|u|2)u= f(|u|2)uin R3,u∈H1(R3,C), $$\begin{array}{} \displaystyle \left\{\!\begin{aligned}&\Big(\frac{\varepsilon {i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert ^{2})u = f(|u|^{2})u\quad\hbox{in }\mathbb{R}^3,\\&u\in H^{1}(\mathbb{R}^ {3}, \mathbb{C}),\end{aligned}\right。\end{array}$$ 其中 ϵ > 0, V : ℝ3 → ℝ 和 A : ℝ3 → ℝ3 是连续势。在对势 V 的局部假设下，通过变分方法、惩罚技术和 Ljusternick-Schnirelmann 理论，我们证明了 ε > 0 小的非平凡解的多重性和集中性。在这个问题中，函数 f 只是连续的，