Lazer-McKenna Conjecture for fractional problems involving critical growth

Abstract

In this paper, the fractional problem of the Ambrosetti-Prodi type involving the critical Sobolev exponent is taken into account in a bounded domain of RN{Aαu=u+2α⁎−1+λu−s¯φ1,u>0, in Ω,u=0, on ∂Ω, where Aα is the spectral fractional operator, λ and s¯ are real numbers, Ω⊂RN is bounded, 2α⁎=2NN−2α is a critical exponent, 0<α<1, φ1 is the first eigenfunction of −Δ with zero Dirichlet boundary condition. We will construct bubbling solutions when the parameter is large enough, and the location of the bubbling point is near the boundary of the domain.