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Standing waves for quasilinear Schrödinger equations with indefinite potentials. (English) Zbl 1404.35130
The paper is focused on the quasilinear stationary Schrödinger equation in \(\mathbb{R}^N\) \[ -\Delta u+V(x)u-u\Delta (|u|^2)=g(u),(1) \] where \(g(u)\) is \(4\)-superlinear, and \(-\Delta+V\) is an indefinite operator. Using Morse theory and the local linking property for an associated functional, the authors prove the existence of a nontrivial solution for \((1)\). Besides, if \(g\) is odd, applying the symmetric mountain pass theorem of Ambrosetti-Rabinowitz, they show the existence of an unbounded sequence of solutions for \((1)\).

MSC:
35J10 Schrödinger operator, Schrödinger equation
35J62 Quasilinear elliptic equations
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