Maximal existence domains of positive solutions for two-parametric systems of elliptic equations

TL;DR

This paper investigates the parameter ranges for which positive solutions exist in systems of elliptic equations with p, q-Laplacians, introducing curves that bound the maximal domain of solutions using variational and Nehari manifold methods.

Contribution

It introduces explicit bounds for the existence domain of positive solutions in two-parametric elliptic systems, extending previous work with new variational techniques.

Findings

Defined the curves  and  as bounds for solution existence domains.

Derived the upper bound curve  explicitly via minimax variational principle.

Extended the applicability of Nehari manifold and fibering methods to these systems.

Abstract

The paper is devoted to the study of two-parametric families of Dirichlet problems for systems of equations with $p, q$-Laplacians and indefinite nonlinearities. Continuous and monotone curves $\Gamma_f$ and $\Gamma_e$ on the parametric plane $\lambda \times \mu$, which are the lower and upper bounds for a maximal domain of existence of weak positive solutions are introduced. The curve $\Gamma_f$ is obtained by developing our previous work \cite{BobkovIlyasov} and it determines a maximal domain of the applicability of the Nehari manifold and fibering methods. The curve $\Gamma_e$ is derived explicitly via minimax variational principle of the extended functional method.