On a positive solution for (p,q)-Laplace equation with indefinite weight
Résumé
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.
