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Bound states for semilinear Schrödinger equations with sign-changing potential. (English) Zbl 1119.35082

Summary: We study the existence and the number of decaying solutions for the semilinear Schrödinger equations \({-\varepsilon^{2}\Delta u + V(x)u = g(x,u)}\), \({\varepsilon > 0}\) small, and \({-\Delta u + \lambda V(x)u = g(x,u)}\), \({\lambda > 0}\) large. The potential \(V\) may change sign and \(g\) is either asymptotically linear or superlinear (but subcritical) in \(u\) as \(|u| \to \infty\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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