Multiple solutions for a fractional p-Kirchhoff problem with Hardy nonlinearity

This paper is devoted to study the existence of multiple solutions for the following fractional p-Kirchhoff problem { M (integral(R2n) vertical bar u(x) - u(y)vertical bar(p)/vertical bar x - y vertical bar(n+ps) dxdy) (-Delta)p( u (0.1) -)(s) lambda vertical bar u vertical bar(q-2 )u + vertical bar u vertical bar(r-2) u/vertical bar x vertical bar(alpha), in Omega, u = 0, in R-n\Omega, where (-Delta)(p)(s )denotes the fractional p-Laplace operator, Omega is a smooth bounded set in a R-n containing 0 with Lipschitz boundary, M(t) = a + bt(theta-1) with a >= 0, b > 0, theta > 1. lambda > 0, 1 < q < p < theta p <= r <= p(alpha)*, p(alpha)*= (n-alpha)p/n-ps is the fractional critical Hardy-Sobolev exponent for 0 <= alpha < ps < n. By using fibering maps and Nehari manifold, we obtain that the existence of multiple solutions to problem 0.1 for both Hardy-Sobolev subcritical and critical cases. In particular, the concentration compactness principle will be used to overcome the lack of compactness for the critical case. (C) 2019 Elsevier Ltd. All rights reserved.