Bifurcations theorems for nonlinear problems with lack of compactness

In this paper we deal with a bifurcation result for the following Dirichlet problem[ left{egin{array}{ll}-Delta_pu= rac{mu}{|x|^p}|u|^{p-2}u+lambda f(x,u)qquad extrm{ a. e. in }Omega\u_{|partialOmega}=0end{array} ight.]Starting from a weak lower semicontinuity result by E. Montefusco, that allows us to apply a general variational principle by B.Ricceri, we prove that, for $mu$ close to zero, there exists apositive number $lambda^*_mu$ such that, for every $lambdain]0,lambda^*_mu[$ the above problem admits a non-zero weaksolution $u_lambda$ in $W_0^{1,p}(Omega)$ satisfying$lim_{lambda o 0^+}|u_lambda|=0$.