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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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