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

TL;DR

This paper explores the existence and multiplicity of positive solutions for a class of $(p,q)$-Laplace equations with two parameters, revealing conditions under which multiple solutions occur near specific critical parameter values.

Contribution

It identifies new regions in parameter space where multiple positive solutions exist, including unexpected cases with three solutions, based on the relation between $p$ and $q$.

Findings

Existence of two positive solutions near certain critical points.

Discovery of conditions leading to three positive solutions.

Analysis of the relation between $p$ and $q$ affecting solution multiplicity.

Abstract

We study the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u+\beta |u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical curves on the $(\alpha,\beta)$-plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighbourhoods of the point $(\alpha,\beta) = \left(\|\nabla \varphi_p\|_p^p/\|\varphi_p\|_p^p, \|\nabla \varphi_p\|_q^q/\|\varphi_p\|_q^q\right)$, where $\varphi_p$ is the first eigenfunction of the $p$-Laplacian, we show the existence of two and, which is rather unexpected, three distinct positive solutions, depending on a relation between the exponents $p$ and $q$.