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

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.

33 pages, 2 figures

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• On the linear independence condition for the Bobkov-Tanaka first eigenvalue of the double-phase operator