We are concerned with the qualitative analysis of solutions for three classes of nonlinear problems driven by the magnetic Laplace operator. We are mainly interested in the perturbation effects created by two reaction terms with different structure. Two equations are studied on bounded domains (under Dirichlet boundary condition) while the third problem is on the entire Euclidean space. Our main results establish that if a certain perturbation is sufficiently small (in a prescribed sense) then the problems have at least two distinct solutions in a related magnetic Sobolev space. The proofs combine variational, topological and analytic methods.

中文翻译：

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具有磁势的非线性Schrödinger方程的小摄动

我们关注由磁性拉普拉斯算子驱动的三类非线性问题的解的定性分析。我们主要对结构不同的两个反应项产生的摄动效应感兴趣。在有界域上（在Dirichlet边界条件下）研究了两个方程，而第三个问题在整个欧几里得空间上。我们的主要结果表明，如果某种扰动足够小（在规定的意义上），则问题在相关的磁Sobolev空间中至少具有两个不同的解决方案。证明结合了变分，拓扑和分析方法。