Existence and multiplicity of positive solutions for fractional Schrödinger equations with critical growth

Abstract

In this paper, we study the existence and multiplicity of positive solutions for the fractional Schrödinger equations ε2α(−Δ)αu+V(x)u=∣u∣2α∗−2u+σg(x,u),x∈RN, where ε and σ are positive parameters, 0<α<1, (−Δ)α denotes the fractional Laplacian of order α, N>2α and 2α∗=2NN−2α is the fractional critical exponent, V∈C(RN,R+), g∈C(RN×R,R). By the variational methods we study the existence and multiplicity of positive solutions. Our main results improve corresponding results in Shang and Zhang (2014).