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Existence of standing waves solution for a nonlinear Schrödinger equation in \(\mathbb{R}^N\). (English) Zbl 1378.35142

Summary: In this paper, we investigate the existence of a positive solution for the following class of elliptic equation \[ -\epsilon^2\Delta u+V(x)u=f(u)\quad\text{in }\mathbb R^N, \] where \(\epsilon > 0\) is a positive parameter, \(f\) has a subcritical growth and \(V\) is a positive potential verifying some conditions.

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J20 Variational methods for second-order elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B09 Positive solutions to PDEs
35B25 Singular perturbations in context of PDEs
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References:

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