Two solutions for a class of singular Kirchhoff-type problems with Hardy-Sobolev critical exponent II

In this article, we devote ourselves to investigate the following singular Kirchhoff-type equation: {-(a+b integral(Omega)vertical bar del u vertical bar(2)dx) Delta u=u5-2s/vertical bar x vertical bar(s) + lambda/vertical bar x vertical bar(beta)w, x is an element of Omega, u>0, x is an element of Omega, u=0, x is an element of delta Omega, where Omega subset of R-3 is a bounded domain with smooth boundary delta Omega, ,0 is an element of Omega, a >= 0, b, lambda > 0, 0 < gamma, s < 1, and 0 <= beta < 5 + gamma/2. By using the variational and perturbation methods, we obtain the existence of two positive solutions, which generalizes and improves the recent results in the literature.