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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