Abstract multiplicity results for $(p,q)$-Laplace equations with two parameters

TL;DR

This paper studies the existence and multiplicity of solutions for a class of $(p,q)$-Laplace equations with two parameters, providing parameter ranges for multiple solutions and new characterizations of eigenvalues.

Contribution

It introduces new parameter ranges for solution multiplicity and offers alternative eigenvalue characterizations for the $q$-Laplacian.

Findings

Identifies three parameter ranges with multiple solutions

Provides new eigenvalue characterizations using constrained variational methods

Establishes solution existence under zero Dirichlet boundary conditions

Abstract

We investigate the existence and multiplicity of abstract weak solutions of the equation $-\Delta_p u -\Delta_q u=\alpha |u|^{p-2}u + \beta |u|^{q-2}u$ in a bounded domain under zero Dirichlet boundary conditions, assuming $1<q<p$ and $\alpha,\beta \in \mathbb{R}$. We determine three generally different ranges of parameters $\alpha$ and $\beta$ for which the problem possesses a given number of distinct pairs of solutions with a prescribed sign of energy. As auxiliary results, which are also of independent interest, we provide alternative characterizations of variational eigenvalues of the $q$-Laplacian using narrower and larger constraint sets than in the standard minimax definition.