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

TL;DR

This paper studies the existence and nonexistence of sign-changing solutions for a two-parameter family of $(p,q)$-Laplace equations, characterizing parameter regions where solutions with specific nodal properties exist.

Contribution

It provides explicit characterization of parameter subsets for nodal solutions using variational methods and explores eigenvalue relations for $p$- and $q$-Laplacians in one dimension.

Findings

Identified parameter regions with nodal solutions

Established conditions for solutions with two nodal domains

Analyzed eigenvalue relations between $p$- and $q$-Laplacians

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 $p \neq q$. 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.