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

TL;DR

This paper investigates the conditions under which positive solutions exist for a class of $(p,q)$-Laplace equations with two parameters, establishing a threshold curve in parameter space and exploring eigenvalue monotonicity in specific domains.

Contribution

It constructs a continuous threshold curve in the $(eta,eta)$ plane for solution existence and provides examples of domains with non-monotone eigenvalues for the $(p,q)$-Laplace operator.

Findings

A continuous curve in $(eta,eta)$ separates existence and non-existence regions.

Examples of domains where the first eigenvalue of $- riangle_p$ is non-monotone in $p$.

Conditions under which positive solutions are guaranteed or ruled out.

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 $p \neq q$, 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. $p > 1$.