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Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1119–1143 | Cite as

Multiple Solutions for a Class of Nonhomogeneous Fractional Schrödinger Equations in \(\mathbb {R}^{N}\)

  • Vincenzo Ambrosio
  • Hichem Hajaiej
Article

Abstract

This paper is concerned with the following fractional Schrödinger equation
$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s} u+u= k(x)f(u)+h(x) \text{ in } \mathbb {R}^{N}\\ u\in H^{s}(\mathbb {R}^{N}), \, u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$
where \(s\in (0,1),N> 2s, (-\Delta )^{s}\) is the fractional Laplacian, k is a bounded positive function, \(h\in L^{2}(\mathbb {R}^{N}), h\not \equiv 0\) is nonnegative and f is either asymptotically linear or superlinear at infinity. By using the s-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that \(|h|_{2}\) is sufficiently small.

Keywords

Fractional Laplacian Mountain pass theorem Extension method Positive solutions 

Mathematics Subject Classification

35A15 35J60 35R11 45G05 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Dipartimento di Scienze Pure e Applicate (DiSPeA)Università degli Studi di Urbino ’Carlo Bo’UrbinoItaly
  2. 2.Department of MathematicsCalifornia State University, Los AngelesLos AngelesUSA

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