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Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions. (English) Zbl 1221.35393

Summary: We consider the magnetic NLS equation
\[ \bigg(\frac{\hbar}{i} \nabla-A(x)\bigg)^2 u+V(x)u- f(|u|^2)u = 0\quad \text{in }\mathbb R^N, \tag{1} \]
where \(N\geq 3\), \(A:\mathbb R^N\to \mathbb R^N\) is a magnetic potential, possibly unbounded, \(V:\mathbb R^N\to \mathbb R\) is a multi-well electric potential, which can vanish somewhere, \(f\) is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution \(u:\mathbb R^N\to \mathbb C\) to (1), under conditions on the nonlinearity which are nearly optimal.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J20 Variational methods for second-order elliptic equations
78A25 Electromagnetic theory (general)
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