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On positive solutions for $$(p,q)$$-Laplace equations with two parameters. (English) Zbl 1328.35052
Summary: We study the existence and non-existence of positive solutions for the $$(p,q)$$-Laplace equation $$-\Delta_pu-\Delta_q u=\alpha|u|^{p-2}u+\beta|u|^{q-2}u$$, where $$p\neq q$$, under the zero Dirichlet boundary condition in $$\Omega$$. The main result of our research is the construction of a continuous curve in $$(\alpha,\beta)$$ plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains $$\Omega$$ for which the corresponding first Dirichlet eigenvalue of $$-\Delta_p$$ is not monotone w.r.t. $$p>1$$.

##### MSC:
 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35B09 Positive solutions to PDEs 35J40 Boundary value problems for higher-order elliptic equations
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