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Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities. (English) Zbl 1283.35122

In this paper, the authors are mainly interested in the following nonlinear Schrödinger equations \[ \begin{aligned} &(-i\epsilon\nabla+A(x))^2w+V(x)w=W(x)g(|w|)w;\\ &(-i\epsilon\nabla+A(x))^2w+V(x)w=W(x)(g(|w|)+|w|^{2^*-2})w\end{aligned} \] with a magnetic field, with both potentials V and W being positive and depending nontrivially on \(x\), and with ground state solutions concentrating on certain sets, which has not been studied much yet in the literature. The most interesting results obtained here refer to the second equation (critical nonlinear problem). They establish the existence and describe the concentration of semiclassical ground states of equation provided either \(\min\, V < \tau_0\) for some \(\tau_0 > 0\), or \(\max\, W > \kappa_0\) for some \(\kappa_0 > 0\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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