CRITICAL GROWTH FRACTIONAL KIRCHHOFF ELLIPTIC PROBLEMS

This article is concerned with the existence and multiplic-ity of positive weak solutions for the following fractional Kirchhoff-Choquard problem: M(& Vert;u & Vert;(2))(-triangle)(s)u=lambda f(x)|u|q-2u+(integral ohm|u(y)|2 & lowast;mu,s|x-y|mu dy)|u|2 & lowast;mu,s-2uin ohm, u >0 in ohm, u= 0 in R-N\ohm, where ohm is open bounded domain of R(N )with C(2 )boundary, N >2 s and s is an element of(0,1), here M models Kirchhoff-type coefficient of the form M(t) =a+bt theta-1, where a,b >0 are given constants. (-triangle)si s fractional Laplace operator, lambda >0 is a real parameter. We explore u sing the variational methods, the existence of solution fo rq is an element of(1,2 & lowast;s)and theta >= 1. Here, 2 & lowast;s=2NN-2sand 2 & lowast;mu,s=2N-mu N-2sis the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality