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

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 and on the parametric plane , which are the lower and upper bounds for a maximal domain of existence of weak positive solutions, are introduced. The curve is obtained by developing our previous work and it determines a maximal domain of the applicability of the Nehari manifold and fibering methods. The curve is derived explicitly via minimax variational principle of the extended functional method.

Keywords:

• system of elliptic equations
• p-laplacian
• indefinite nonlinearity
• Nehari manifold
• fibering method

AMS Subject Classifications:

• 35J50
• 35J55
• 35J60
• 35J70
• 35R05

Acknowledgements

We thank the referee for bringing [22,23] to our attention.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was partly supported by RFBR [grant number 13-01-00294-p-a]. The second author was partly supported by RFBR [grant number 14-01-00736-p-a].