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1 Introduction
The paper deals with the following fractional Hardy-Sobolev equation with nonhomogeneous term being O < s < 1, where Q is a bounded domain in Rw, (TV >2s) containing the origin 0 in its interior, positive parameter, 0 < a < 2s, 2*s = n^2s is the fractional critical Hardy-Sobolev exponent. The fractional Laplacian (-A)s is defined by where
T is the Gamma function, ƒ is a given bounded measurable function. It has been seen that the fractional differential equations have better effects in many realistic applications than the classical ones. Qualitative theory and its ap-plications in physics, engineering, economics, biology, and ecology are extensively discussed and demonstrated in [5, 6, 8, 11, 12, 13] and the references therein. There have been by now a large number of papers concerning the existence, nonex-istence as well as qualitative properties of nontrivial solutions to critical elliptic problems of Hardy potential and fractional Laplace operator. For instance, Ben-nour and all in [1] handled the following singular equation
where Q is a bounded domain in R and N >5,Under sufficiënt conditions on the data, the existence and multiplicity of solutions was proven, via Ekland's variation principle and the Mountain Pass Lemma principle.
In the local setting case (s = 1) the problem (1) is reduced to the semilinear problem with Sobolev-Hardy exponents
This problem was further studied by Chen and Rocha [4], who based on varia-tional methods obtained the existence of four non-trivial solutions. Recently, the existence of nontrivial solutions for nonlinear fractional elliptic equations with Hardy's potential type have been studied by several authors. Wang and all [15] studied (2) with g(u) = \u\2*~2u + au, a >0 and discussed the infinitely many solutions. Daoues and all
[7] studied (2) with g(u) = X\u\q~2u + |M| ,,t- and obtained the existence and nonexistence of nonnegative distributional solutions.
In what...