Three solutions for a fractional Schrödinger equation with vanishing potentials

Abstract In this paper, we study the following fractional Schrodinger equation ( − Δ ) s u + V ( x ) u = K ( x ) f ( u ) + λ W ( x ) | u | p − 2 u , x ∈ R N , where λ > 0 is a parameter, ( − Δ ) s denotes the fractional Laplacian of order s ∈ ( 0 , 1 ) , N > 2 s , W ∈ L 2 2 − p ( R N , R + ) , 1 p 2 , V , K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. Under some mild assumptions, we prove that the above equation has three solutions.

[1]  Xiaojun Chang,et al.  Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian , 2014 .

[2]  N. Laskin.  Fractional Schrödinger equation. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  L. Caffarelli,et al.  An Extension Problem Related to the Fractional Laplacian , 2006, math/0608640.

[4]  C. O. Alves,et al.  Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity , 2013 .

[5]  A. Ambrosetti,et al.  Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity , 2005 .

[6]  A. Quaas,et al.  Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian , 2012, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[7]  I. Ekeland.  On the variational principle , 1974 .

[8]  Tatsuya Miura.  Singular perturbation by bending for an adhesive obstacle problem , 2015, 1502.04212.

[9]  Xian Wu,et al.  Existence of Positive Solutions for a Class of Critical Fractional Schrödinger Equations with Potential Vanishing at Infinity , 2017, Mediterranean Journal of Mathematics.

[10]  Nicoletta Del Buono,et al.  A Dynamical System Approach for Continuous Nonnegative Matrix Factorization , 2017 .

[11]  N. Laskin.  Fractional quantum mechanics and Lévy path integrals , 1999, hep-ph/9910419.

[12]  C. O. Alves,et al.  Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^N$ via penalization method , 2015, 1508.03964.

[13]  M. Squassina,et al.  CRITICAL AND SUBCRITICAL FRACTIONAL PROBLEMS WITH VANISHING POTENTIALS , 2014, 1410.0843.

[14]  E. Valdinoci,et al.  Hitchhiker's guide to the fractional Sobolev spaces , 2011, 1104.4345.