On sign-changing solutions for $(p,q)$-Laplace equations with two parameters

Abstract

We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for two-parametric family of partially homogeneous $(p,q)$-Laplace equations $-{\Delta }_{p}u-{\Delta }_{q}u=\alpha |u{|}^{p-2}u+\beta |u{|}^{q-2}u$ where $p\ne q$. By virtue of the Nehari manifolds, linking theorem, and descending flow, we explicitly characterize subsets of $(\alpha ,\beta )$-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.

32 pages, 1 figure; minor text improvements performed. To appear in Advances in Nonlinear Analysis

References in corpus (1)

• Variational properties of the first curve of the Fuč\'ık spectrum for elliptic operators

Cited by in corpus (3)

• Abstract multiplicity results for $(p,q)$-Laplace equations with two parameters
• On the Fredholm-type theorems and sign properties of solutions for $(p,q)$-Laplace equations with two parameters
• Multiplicity of positive solutions for $(p,q)$-Laplace equations with two parameters