Elliptic equations involving supercritical Sobolev growth with mixed Dirichlet-Neumann boundary conditions

Abstract

This paper concerns elliptic problems involving supercritical Sobolev growth without the (AR) condition and with a mixed boundary Dirichlet-Neumann type condition. The conditions imposed on the nonlinearity considered here generalizes several previous papers, including that presented in the work that inspired this paper, due to Peral and Colorado, in 2003. Beyond that, we present some complementary results, concerning the non-existence of solutions to a class of elliptic problems and a comparison result inspired by the case of Dirichlet boundary conditions, presented by the work of Ambrosetti, Brezis and Cerami.

Keywords:

• Galerkin method
• elliptic Dirichlet-Neumann equations
• supercritical growth

AMS Subject Classifications:

• MSC 35B09
• 35B33
• 35J25

Acknowledgments

The authors are grateful to the referees for their valuable suggestions that have improved this article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Luiz F.O Faria was partially supported by CNPQ and FAPEMIG CEX APQ 02374/17, and Heitor R. de Assis was partially supported by BIC/UFJF - 2019 PBPG/UFJF - 2021.