Existence of solutions for fractional Schrodinger equation with asymptotica- lly periodic terms

Abstract

In this paper, we investigate the following nonlinear fractional Schrödinger equationwhere s ∈ (0, 1), N > 2 and (−∆) s is fractional Laplacian operator. We prove that the problem has a non-trivial solution under asymptotically periodic case of V and f at infinity. Moreover, the nonlinear term f does not satisfy any monotone condition and Ambrosetti-Rabinowitz condition.

Energy solutions and concentration problem of fractional Schrödinger equation

“…By using a new function space introduced in [38,39] and constructing some inequalities, we can obtain the concentration of the solutions of (1.1) under different conditions. Hence our results can be viewed as an extension to the main results in [16][17][18][19][20][21][22][23]38]. We list the following assumptions.…”

“…In [22], the authors also studied the existence of infinitely many nontrivial energy solutions by variational methods. In [23], in the asymptotically periodic case, a nontrivial solution is obtained by variational methods. For more related study, the interested reader may consult [24][25][26][27][28][29][30][31][32][33][34] and the references therein.…”

Least energy sign-changing solutions for the fractional Schrödinger–Poisson systems in R 3 $\mathbb{R}^{3}$

“…There are many interesting papers which considered the existence, multiplicity, uniqueness, regularity and asymptotic decay properties of the solutions to fractional Schrödinger equation (13), see [1,5,12,18,22,35,39,40,46,59] and references therein. Besides, some more complicated fractional equations and systems were also studied, and indeed some interesting results were obtained, see [19,45,[53][54][55] and references therein.…”