A Brezis–Nirenberg splitting approach for nonlocal fractional equations. (June 2015) - Molica Bisci, Giovanni; Servadei, Raffaella
Abstract: In this paper we consider problems modeled by the following nonlocal fractional equation { ( − Δ ) s u + a ( x ) u = μ f ( u ) in  Ω u = 0 in  R n ∖ Ω, where s ∈ ( 0, 1 ) is fixed, Ω is an open bounded subset of R n, n > 2 s, with Lipschitz boundary, ( − Δ ) s is the fractional Laplace operator and μ is a real parameter. Under two different types of conditions on the functions  a and f, 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.