Generalized Picone inequalities and their applications to $(p,q)$-Laplace equations

We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = f_\mu(x,u,\nabla u)$ in a bounded domain $\Omega \subset \mathbb{R}^N$ under certain assumptions on the nonlinearity and with a special attention to the resonance case $f_\mu(x,u,\nabla u) = \lambda_1(p) |u|^{p-2} u + \mu |u|^{q-2} u$, where $\lambda_1(p)$ is the first eigenvalue of the $p$-Laplacian.