ON A NONLOCAL p(x)-LAPLACIAN DIRICHLET PROBLEM INVOLVING SEVERAL CRITICAL SOBOLEV-HARDY EXPONENTS

The aim of this work is to present a result of multiplicity of solutions, in generalized Sobolev spaces, for a non-local elliptic problem with p(x)-Laplace operator containing k distinct critical Sobolev-Hardy exponents combined with singularity points {M(psi(u))(-triangle(p(x))u+|u|p((x)-2)u) =& sum;(k)(i=1) h(i)(x)|u|p(& lowast;s)i(x)(-2)u/|x|(si)(x)+f(x,u) in ohm, {u= 0 on partial derivative ohm, where ohm subset of R-N is a bounded domain with 0 is an element of ohm and 1< p-<= p(x)<= p+< N. The real function M is bounded in[0,+infinity)and the functions hi(i= 1,...,k)and fare functions whose properties will be given later. To obtain the result we use the Lions' concentration-compactness principle for critical Sobolev-Hardy exponent in the space W-0(1,p(x))(ohm)due to Yu, Fu and Li, and the Fountain Theorem.