Electronic Journal of Differential Equations, Vol. 2010(2010), No. 26, pp. 1-10. Title: Existence and uniqueness for a p-Laplacian nonlinear eigenvalue problem Authors: Giovanni Franzina (Univ. of Trento, Italy) Pier Domenico Lamberti (Univ. of Padova, Italy) Abstract: We consider the Dirichlet eigenvalue problem $$ -\mathop{\rm div}(|\nabla u|^{p-2}\nabla u ) =\lambda \| u\|_q^{p-q}|u|^{q-2}u, $$ where the unknowns $u\in W^{1,p}_0(\Omega )$ (the eigenfunction) and $\lambda >0$ (the eigenvalue), $\Omega $ is an arbitrary domain in $\mathbb{R}^N$ with finite measure, $1