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Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. (English) Zbl 1270.35218

Summary: In the present paper, we study the existence of solutions for a nonlocal Schrödinger equation \[ -\varepsilon^2 \Delta u + V(x)u = \left(\int_{\mathbb R^3} \frac{|u|^{p}}{|x - y|^{\mu}}dy\right)|u|^{p-2}u, \] where \(0 < \mu < 3\) and \(\frac{6 - \mu}{3} < p < {6 - \mu}\). Under suitable assumptions on the potential \(V(x)\), if the parameter \(\varepsilon\) is small enough, we prove the existence of solutions by using the mountain-pass theorem.

MSC:

35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35J75 Singular elliptic equations
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