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On sign-changing solutions for \((p,q)\)-Laplace equations with two parameters. (English) Zbl 1419.35071
Summary: We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous \((p,q)\)-Laplace equations \(-\Delta_{p}u-\Delta_{q}u=\alpha\vert u\vert^{p-2}u+\beta\vert u\vert^{q-2}u\) where \(p\neq q\). By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the \((\alpha,\beta)\)-plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the \(p\)- and \(q\)-Laplacians in one dimension.

MSC:
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35A15 Variational methods applied to PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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