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Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. (English) Zbl 1090.35077

Based on new information concerning strongly indefinite functionals without Palais-Smale conditions, we study existence and multiplicity of solutions of the Schrödinger equation \[ -\Delta u+V(x)u=g(x,u)\quad\text{for }x\in\mathbb R^N,\quad u(x)\to 0\quad \text{as }|x|\to\infty, \] where \(V\) and \(g\) are periodic with respect to \(x\) and 0 lies in a gap of \(\sigma(-\Delta+V)\). Supposing \(g\) is asymptotically linear as \(|u|\to \infty\) and symmetric in \(u\), we obtain infinitely many geometrically distinct solutions. We also consider the situation where \(g\) is superlinear with mild assumptions different from those studied previously, and establish the existence and multiplicity.

MSC:

35J60 Nonlinear elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
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