Fractional Schrödinger equations with new conditions

Abstract

In this article, we study the nonlinear fractional Schrödinger equation (-∆)αu + V(x)u = ƒ(x, u) u ∈ Hα (ℝn, ℝ), where (-∆)α(α ∈ (0, 1)) stands for the fractional Laplacian of order α, x ∈ ℝn, V ∈ C(ℝn, ℝ) may change sign and ƒ is only locally defined near the origin with respect to u. Under some new assumptions on V and ƒ, we show that the above system has infinitely many solutions near the origin. Some examples are also given to illustrate our main theoretical result.