Positive solutions for Dirichlet problems of singular quasilinear elliptic equations via variational methods

This paper studies the existence and multiplicity of positive solutions of the following problem:{-Delta pu = u beta delta gamma(x) + lambda a(x) u alpha, in Omega {u 0, in Omega, (*) {u = 0, on partial derivative Omega,where Omega subset of R-N(N = 3) is a smooth bounded domain, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), 1 p N, and 0 alpha 1, p-1 beta p* -1 (p* = Np (N - p)) and 0 gamma N + ((beta + 1)(p - N) p) are three constants. Also delta(x) = dist(x, partial derivative Omega), a is an element of L-p and lambda 0 is a real parameter. By using the direct method of the calculus of variations, Ekelands Variational Principle and an idea of G. Tarantello, it is proved that problem (*) has at least two positive weak solutions if I is small enough.