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Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials. (English) Zbl 1131.35075

Summary: We establish the existence and multiplicity of semiclassical bound states of the following nonlinear Schrödinger equation:
\[ \begin{cases} -\varepsilon^2\Delta u+V(x)u=g(x,u) \quad & \text{for }x\in\mathbb R^N,\\ u(x)\to\infty & \text{as }|x|\to\infty\end{cases} \]
where \(V\) changes sign and \(g\) is superlinear with critical or supercritical growth as \(|u|\to\infty\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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