Generalized critical Kirchhoff-type potential systems with Neumann Boundary conditions

Abstract

In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems with Neumann Boundary conditions, which involves a general variable exponent elliptic operator with critical growth. Under some suitable conditions on the nonlinearities, we establish existence and multiplicity of solutions for the problem by using the concentration-compactness principle of Lions for variable exponents and the mountain pass theorem without the Palais–Smale condition.

Keywords:

• Variable exponent spaces
• critical Sobolev exponents
• Kirchhoff-type problems
• p-Laplcian
• $p(x)$-Laplacian
• generalized Capillary operator
• Neumann Boundary conditions
• concentration-compactness principle
• Palais–Smale condition
• mountain pass theorem
• critical points theory

2010 Mathematics Subject Classifications:

• 35B33
• 35D30
• 35J50
• 35J60
• 46E35

Acknowledgements

The second author wish to thank GNAMPA and the RUDN University Strategic Academic Leadership Program.

Disclosure statement

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

Additional information

Funding

This work was supported by GNAMPA [grant number 2020], PRIN 2017 n.2017AYM8XW004 [grant number 2017AYM8XW004] and RUDN University Strategic Academic Leadership Program [grant number 5-100].