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This paper is devoted to the following class of nonlinear fractional Schrödinger equations: (−∆) s u + V (x)u = f (x, u) + λg(x, u), in R N , where s ∈ (0, 1), N > 2s, (−∆) s stands for the fractional Laplacian, λ ∈ R is a parameter, V ∈ C(R N , R), f (x, u) is superlinear and g(x, u) is sublinear with respect to u, respectively. We prove the existence of infinitely many high energy solutions of the aforementioned equation by means of the Fountain theorem. Some recent results are extended and sharply improved.

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In this article we study the problemwhere 2 := ( ) is the biharmonic operator, a, b > 0 are constants, N ≤ 7, p ∈ (4, 2 * ) for 2 * defined below, and V (x) ∈ C(R N , R). Under appropriate assumptions on V (x), the existence of least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.

This paper is devoted to study a class of nonlinear fractional Schrödinger equations:where s ∈ (0, 1), N > 2s, (−∆) s stands for the fractional Laplacian. The main purpose of this paper is to study the existence of infinitely many solutions for the aforementioned equation. By using variational methods and the genus properties in critical point theory, we establish the existence of at least one nontrivial solution as well as infinitely many solutions for the above equation with a general potential V (x) which is allowed to be sign-changing and the nonlinearity f (x, u) is locally sublinear with respect to u.