Multiplicity of solutions for some singular quasilinear Schrodinger-Kirchhoff equations with critical exponents

In this paper, we focus on the existence of multiplicity of solutions for the singular quasilnear Schrodinger-Kirchhoff type problem: - (a + b integral(R3) (1 + alpha(2)/2 vertical bar u vertical bar(2(alpha-1)) ) vertical bar del u vertical bar(2) dx) (Delta u + alpha/2 Delta(vertical bar u vertical bar(alpha))vertical bar u vertical bar(alpha-2)u) = lambda V(x)vertical bar u vertical bar(p-2)u + G(x)vertical bar u vertical bar(4)u, where x is an element of R-3, a> 0, b >= 0, 0 < alpha < 1, 1 < p < 2, lambda is an element of R, 0 <= V(x) is an element of C(R-3) boolean AND L-q(R-3) with q = 6/6-p, G(x) is an element of C(R-3) boolean AND L-infinity(R-3), and we get the existence of infinitely many weak solutions by variational methods.