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 $-Δ_p u -Δ_q u = α|u|^{p-2} u + β|u|^{q-2} u$, where $p \neq q$, under the zero Dirichlet boundary condition in $Ω$. The main result of our research is the construction of a continuous curve in $(α,β)$ plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains $Ω$ for which the corresponding first Dirichlet eigenvalue of $-Δ_p$ is not monotone w.r.t. $p > 1$.

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