Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling

Abstract

It is well known that a single nonlinear fractional Schrödinger equation with a potential $V(x)$ and a small parameter $\epsilon$ may have a positive solution that is concentrated at the nondegenerate minimum point of $V(x)$. In this paper, we can find two different positive solutions for two weakly coupled fractional Schrödinger systems with a small parameter $\epsilon$ and two potentials $V_1(x)$ and $V_2(x)$ having the same minimum point are concentrated at the same point minimum point of $V_1(x)$ and $V_2\left(x\right)$. In fact that by using the energy estimates, Nehari manifold technique and the Lusternik-Schnirelmann theory of critical points, we obtain the multiplicity results for a class of fractional Laplacian system. Furthermore, the existence and nonexistence of least energy positive solutions are also explored.

33pages