LAUSR

Abstract In this paper, we study the following class of nonlinear boundary value problems (P)a { − L K u = f ( x , u ) + H ( u − a ) | u | 2 s ⁎ − 2 u  in  Ω , u ≥ 0  in  Ω , u = 0  in  R N ∖ Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, N > 2 s , s ∈ ( 0 , 1 ) , 2 s ⁎ = 2 N / ( N − 2 s ) , is a fractional critical Sobolev exponent, a ≥ 0 is a real parameter, H is the Heaviside function, i.e., H ( t ) = 0 if t ≤ 0 , H ( t ) = 1 if t > 0 , and L K is the nonlocal operator L K u ( x ) : = ∫ R N ( u ( x + y ) + u ( x − y ) − 2 u ( x ) ) K ( y ) d y , for x ∈ R N . We prove, using appropriate hypotheses on f and K, existence of solutions u a for ( P ) a and then we prove that such sequence converges, in a certain sense, to a solution of the problem ( P ) 0 as a → 0 + .