
Minimax Theory and its Applications 01 (2016), No. 1, 001020 Copyright 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@univperp.fr Mieko Tanaka Department of Mathematics, Tokyo University of Science, Kagurazaka 13, Shinjyukuku, Tokyo 1628601, Japan tanaka@ma.kagu.tus.ac.jp [Abstractpdf] This paper provides existence and nonexistence results for a positive solution of the quasilinear elliptic equation $$ \Delta_p u\mu\Delta_q u = \lambda (m_p(x)u^{p2}u+\mu m_q(x)u^{q2}u) \quad {\rm in}\ \Omega $$ driven by the nonhomogeneous operator $(p,q)$Laplacian under Dirichlet boundary condition, with $\mu>0$ and $1 0$ 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 nonexistence result is also given. 