Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Minimax Theory and its Applications 01 (2016), No. 1, 001--020Copyright Heldermann Verlag 2016 On a Positive Solution for (p,q)-Laplace Equation with Indefinite Weight Dumitru Motreanu Départment de Mathématiques, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France motreanu@univ-perp.fr Mieko Tanaka Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjyuku-ku, Tokyo 162-8601, Japan tanaka@ma.kagu.tus.ac.jp [Abstract-pdf] This paper provides existence and non-existence results for a positive solution of the quasilinear elliptic equation $$-\Delta_p u-\mu\Delta_q u = \lambda (m_p(x)|u|^{p-2}u+\mu m_q(x)|u|^{q-2}u) \quad {\rm in}\ \Omega$$ driven by the nonhomogeneous operator $(p,q)$-Laplacian under Dirichlet boundary condition, with $\mu>0$ and $10$ the results are completely different from those for the usual eigenvalue problem for the $p$-Laplacian, which is retrieved when $\mu=0$. For instance, we prove that when $\mu>0$ there exists an interval of eigenvalues. Existence of positive solutions is obtained in resonant cases, too. A non-existence result is also given. Keywords: (p,q)-Laplacian, nonlinear eigenvalue problems, indefinite weight, mountain pass theorem, global minimizer. MSC: 35J62, 35J20, 35P30 [ Fulltext-pdf  (190  KB)] for subscribers only.