On a nonlocal nonhomogeneous Neumann boundary problem with two critical exponents

ABSTRACT

In this paper, we are concerned with questions of the existence of solution for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the $p\left(x\right)$-Laplacian in which the nonlinear terms assume both critical growth. The main tools used are the Lions' Concentration-Compactness Principle [Lions PL. The concentration-compactness principle in the calculus of variations. The limit case I, part 1. Rev Mat Iberoam. 1985;1(1):145–201.] for variable exponent spaces and the Mountain Pass Theorem.

KEYWORDS:

• Nonlocal problem
• Neumann boundary conditions
• trace
• sobolev spaces with variable exponent
• critical exponent

AMS SUBJECT CLASSIFICATIONS:

• 35J60
• 35J70
• 58E05

Acknowledgments

This paper was done while the first author was visiting the Department of Mathematics of the Federal University of Juiz de Fora, performing stage of post-doctoral, whose hospitality and support he gratefully acknowledge. The authors also gratefully acknowledge the constructive comments made by referees.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

L. S. Tavares  http://orcid.org/0000-0003-0953-8037

Additional information

Funding

This work was financially supported by PNPD/CAPES [2017-PGM/UFJF].