Existence of at least k solutions to a fractional p-Kirchhoff problem involving singularity and critical exponent

We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity M(integral(Q)|u(x)-u(y)|(p) / |x-y|(N+ps) dxdy)(-Delta)(p)(s)u=lambda/u(gamma)+u(ps & lowast;)-1 in Omega, u>0 in Omega, u=0 in R-N\Omega, where M is the Kirchhoff function, Q=R-2N\((R-N\Omega)x(R-N\Omega)), Omega subset of R-N, is a bounded domain with Lipschitz boundary, lambda>0, N>ps, 0<s,gamma<1, (-Delta)(p)(s) is the fractional p-Laplacian for 1<p<infinity and p(s)(& lowast;)=Np/N-ps is the critical Sobolev exponent. We employ a cut-off argument to obtain the existence of k (being arbitrarily large integer) solutions. Furthermore, by using the Moser iteration technique, we prove an uniform L-infinity(Omega) bound for the solutions. The novelty of this work lies in proving the existence of small energy solutions by using symmetric mountain pass theorem in spite of the presence of a critical nonlinear term which, of course, is super-linear.