The Neumann problem for a class of generalized Kirchhoff-type potential systems

Abstract

Abstract In this paper, we are concerned with the Neumann problem for a class of quasilinear stationary Kirchhoff-type potential systems, which involves general variable exponents elliptic operators with critical growth and real positive parameter. We show that the problem has at least one solution, which converges to zero in the norm of the space as the real positive parameter tends to infinity, via combining the truncation technique, variational method, and the concentration–compactness principle for variable exponent under suitable assumptions on the nonlinearities.

Keywords

• Kirchhoff-type problems
• Neumann boundary conditions
• p ( x ) $p(x)$ -Laplacian operator
• Generalized capillary operator
• Sobolev spaces with variable exponent
• Critical Sobolev exponents