Infinitely many solutions of the nonlinear fractional Schrödinger equations

Multiplicity results for a fractional Schrodinger equation with potentials

“…(2) Under the condition (V ) it is clear that V (x) is sign-changing and not satisfied any coerciveness condition, so the main difficulties of our problem is the lack of compactness of the Sobolev embedding. Noting that in [13,18,19] the authors studied problem (1.1) with a potential function V (x) which is strictly positive, so our results extend and improve the aforementioned works.…”

“…(1) Unlike [13,18,19], the nonlinear term f does not satisfy any growth condition and any control at infinity, and we just require f (x, u) locally odd with respect to u when we prove the existence of infinitely many solutions. Moreover, there are functions f (x, u) satisfying (F 1 ) − (F 3 ), for example, let…”

Existence of infinitely many high energy solutions for a fractional Schrödinger equation in RN

“…In particular, the existence of infinitely many high or small energy solutions to problem (1.1) was established in [5,7,9,11,12,13,17,19] by the aid of variant fountain theorems (see [24]) or the symmetric mountain pass theorem (see [23]). However, there are few papers concern with the existence of infinitely many (high or small) energy solutions to problem (1.1) in the case where f (x, u) is a combination of sublinear and superlinear terms at infinity with respect to u, see for instance [7,17,20].…”

“…In [7], Du and Tian considered the following class of fractional Schrödinger equation with concave and critical nonlinearities…”

“…In recent years, the equation (1.1) has been extensively studied under various assumptions on V and f and there are many interesting results in the literature on the existence and multiplicity of solutions to problem (1.5) has been obtained via variational approaches, we refer the readers to [3,5,7,8,9,11,12,13,17,18,19,20,21,22]. In particular, the existence of infinitely many high or small energy solutions to problem (1.1) was established in [5,7,9,11,12,13,17,19] by the aid of variant fountain theorems (see [24]) or the symmetric mountain pass theorem (see [23]). However, there are few papers concern with the existence of infinitely many (high or small) energy solutions to problem (1.1) in the case where f (x, u) is a combination of sublinear and superlinear terms at infinity with respect to u, see for instance [7,17,20].…”