MULTIPLE SOLUTIONS OF FRACTIONAL KIRCHHOFF EQUATIONS INVOLVING A CRITICAL NONLINEARITY

In this paper, we are concerned with the following fractional Kirchhoff equation { (a + b integral N-R vertical bar (-Delta)s/2 u vertical bar(2)) (-Delta)(s)(u) = lambda u + mu vertical bar u vertical bar(q-2) u + vertical bar u vertical bar(s)(2)*(-2)u in Omega, R-N\Omega in u=0 where N > 2s, a,b,lambda,mu > 0, s is an element of (0,1) and Omega is a boundeden domain with continuous boundary. Here (-Delta)(s) is the fractional Laplacian operator. For 2 < q <= min {4,2(S)(*)} we prove that if b is small or is large, the problem above admits multiple solutions by virtue of a linking theorem due to G. Cerami, D. Fortunato and M. Struwe [7, Theorem 2.5].