A Brezis–Nirenberg splitting approach for nonlocal fractional equations

In this paper we consider problems modeled by the following nonlocal fractional equation {(−Δ)su+a(x)u=μf(u)in  Ωu=0in  Rn∖Ω, where s∈(0,1)s∈(0,1) is fixed, ΩΩ is an open bounded subset of RnRn, n>2sn>2s, with Lipschitz boundary, (−Δ)s(−Δ)s is the fractional Laplace operator and μμ is a real parameter.Under two different types of conditions on the functions  aa and ff, by using a famous critical point theorem in the presence of splitting established by Brezis and Nirenberg, we obtain the existence of at least two nontrivial weak solutions for our problem.