Nonlocal Hénon type problem with nonlinearities involving slightly subcritical growth

In this paper, we study the existence of solutions for the following nonlocal superlinear elliptic problem (0.1) { ( − Δ ) s u = β ( x ) u p − ε in Ω , u = 0 in R n ﹨ Ω , where s ∈ ( 0 , 1 ) , n > 2 s , p : = n + 2 s n − 2 s is the Sobolev critical exponent, Ω ⊂ R n is a smooth bounded domain with Lipschitz boundary, ( − Δ ) s is the fractional Laplace operator and β ∈ C 2 ( Ω ‾ ) is a bounded positive continuous function. We assume that there exists a nondegenerate critical point ξ ⁎ ∈ ∂ Ω of the restriction of β to the boundary ∂Ω such that ∇ ( β ( ξ ⁎ ) − n − 2 s 2 s ) ⋅ η ( ξ ⁎ ) > 0 . Given any integer k ≥ 1 , we show that for ε > 0 small enough, problem (0.1) has a positive solution, which is a sum of k bubbles which concentrate at ξ ⁎ as ε tends to zero. Also, we prove the existence of nodal (sign changing) solution whose shape resembles a sum of a positive bubble and a negative bubble near the point ξ ⁎ . This work can be seen as a nonlocal analogue of the result by Dávila, Faya and Mahmoudi, see [28] .