Some remarks on the solvability of non-local elliptic problems with the Hardy potential

The aim of this paper is to study the solvability of the following problem, [GRAPHICS] where (-Delta)(s), with s is an element of (0, 1), is a fractional power of the positive operator -Delta, Omega subset of R-N, N > 2s, is a Lipschitz bounded domain such that 0 is an element of Omega, mu is a positive real number, lambda < A(N,s), the sharp constant of the Hardy-Sobolev inequality, 0 < q < 1 and 1 < p < p(lambda, s) equivalent to N+2s-2 alpha(lambda)/N-2s-2 alpha(lambda), with alpha(lambda) a parameter depending on lambda and satisfying alpha(lambda) is an element of (0, N-2s/2). We will discuss the existence and multiplicity of solutions depending on the value of p, proving in particular that p(lambda, s) is the threshold for the existence of solution to problem (P-mu).