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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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References:
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