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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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