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arXiv · 1706.03034

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

Vladimir Bobkov·Mieko Tanaka
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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 $-Δ_p u -Δ_q u = α|u|^{p-2}u + β|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$ under zero Dirichlet boundary condition, where $p > q > 1$ and $α, β\in \mathbb{R}$. A curve on the $(α,β)$-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.

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Vladimir Bobkov, Mieko Tanaka. 2017-06-09. Remarks on minimizers for $(p,q)$-Laplace equations with two parameters. https://doi.org/10.3934/cpaa.2018059

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