# Remarks on minimizers for $(p,q)$-Laplace equations with two parameters

## TL;DR

This paper investigates the existence, nonexistence, and properties of solutions to a two-parameter $(p,q)$-Laplace equation, constructing an existence curve and analyzing eigenfunction independence.

## Contribution

It provides a detailed analysis of solution existence and multiplicity for the $(p,q)$-Laplace equation, including the construction of an existence curve in the parameter space.

## Key findings

- Existence and nonexistence regions for solutions are characterized.
- A curve in the $(oldsymbol{ ext{α}}, oldsymbol{ ext{β}})$-plane is constructed.
- Eigenfunctions of $p$- and $q$-Laplacians are shown to be linearly independent.

## Abstract

We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the $(p,q)$-Laplace equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u + \beta |u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$ under zero Dirichlet boundary condition, where $p > q > 1$ and $\alpha, \beta \in \mathbb{R}$. A curve on the $(\alpha,\beta)$-plane which allocates a set of the existence of ground states and the multiplicity of positive solutions is constructed. Additionally, we show that eigenfunctions of the $p$- and $q$-Laplacians under zero Dirichlet boundary condition are linearly independent.