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We deal with a bifurcation result for the Dirichlet problem ⎧$-{\Delta }_{p}u=\mu /{\left|x\right|}^p{\left|u\right|}^{p-2}u+\lambda f(x,u)$ a.e. in Ω, ⎨ ⎩$u_{|\partial \Omega }=0$. Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number $\lambda {*}_{\mu }$ such that for every $\lambda \in ]0,\lambda {*}_{\mu }[$ the above problem admits a nonzero weak solution $u_{\lambda }$ in $W{₀}^{1,p}\left(\Omega \right)$ satisfying $li{m}_{\lambda \to 0⁺}\left|\right|u_{\lambda }\left|\right|=0$.