Non-homogeneous Fractional Laplacian Involving Critical Sobolev Exponent

In this paper, we study the existence of solutions to the following critical non-homogeneous elliptic problem involving the fractional Laplacian: Where Q C (?)RN is a smooth bounded domain, N≥1,0<α< min{N,2},f is a nonnegative function, λ∈R R is a fixed constant, and μ>0 is a parameter, p=2α*- 1 and 2α*=2N/N-α denoted the fractional critical Sobolev exponent. We investigate the multiplicity of positive solutions to the problem and find the phenomenon depending on the space dimension N. Precisely, we show that the situation is different between the case α<N<3α and N>3α if λ>0.