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Minimax Theory and its Applications 01 (2016), No. 1, 001--020
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

Mieko Tanaka
Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjyuku-ku, Tokyo 162-8601, Japan


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

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