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

Abstract

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 $p\ne q$, under the zero Dirichlet boundary condition in $\Omega$. 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 $\Omega$ for which the corresponding first Dirichlet eigenvalue of $-{\Delta }_p$ is not monotone w.r.t. $p>1$.

28 pages, 3 figures

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