On positive solutions for $(p,q)$-Laplace equations with two parameters

We study the existence and non-existence of positive solutions for the $(p,q)$-Laplace equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2} u + \beta |u|^{q-2} u$,where$p \neq q$,underthezeroDirichletboundaryconditionin$\Omega$. The main result of our research is the construction of a continuous curve in $(\alpha ,\beta )$ plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains $\mathrm{\Omega }$ for which the corresponding first Dirichlet eigenvalue of $-{\mathrm{\Delta }}_p$ is not monotone w.r.t. $p>1$.