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Remarks on minimizers for \((p,q)\)-Laplace equations with two parameters. (English) Zbl 1400.35145
Summary: 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.

MSC:
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J40 Boundary value problems for higher-order elliptic equations
35J35 Variational methods for higher-order elliptic equations
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