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Solutions of nonlinear Schrödinger equation with fractional Laplacian without the Ambrosetti-Rabinowitz condition. (English) Zbl 1338.35463
Summary: This paper is concerned with the existence of two nonnegative radial solutions of following nonlinear Schrödinger equation with fractional Laplacian \[ (- \Delta)^\alpha u + u = f(u) \text{ in } \mathbb{R}^N, \quad u \in H^\alpha(\mathbb{R}^N), \] where \(0 < \alpha < 1\). Under certain assumptions, we obtain that the above problem has at least two nontrivial radial solutions without assuming the Ambrosetti-Rabinowitz condition by variational methods and concentration compactness principle. The result extends one of the main results of P. Felmer et al. [Proc. R. Soc. Edinb., Sect. A, Math. 142, No. 6, 1237–1262 (2012; Zbl 1290.35308)].

MSC:
35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
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