On multiplicity of positive solutions for N-Laplacian with singular and critical nonlinearity

In this note, we study the existence of multiple positive solutions to (P-lambda) {-del center dot (vertical bar del u vertical bar(N-2)del u) = lambda(a(x) u(-eta) + m(x)u(q-1)) + h(x, u) e vertical bar u vertical bar(N/N-1) in Omega u = 0 on partial derivative Omega where lambda > 0, N >= 2, eta > 0, 1 < q < N, m >= 0, parallel to m parallel to (L)infinity = 1 and Omega is a smooth bounded domain in IRN. Under suitable assumptions on a(x) and h(x, u), we show that there exists Lambda > 0 such that (P-lambda) admits two solutions for lambda is an element of (0, Lambda) and no solution for lambda > Lambda.