On Nonlocal Elliptic Systems of $p(x)$-Kirchhoff-Type under Neumann Boundary Condition

Abstract:

      This paper is concerned with the existence of solutions to a class of $p(x)$-Kirchhoff-type systems under Neumann boundary condition. By Ekeland Variational Principle and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence of solutions for the problem. Since the Poincar\'{e}'s inequality does not hold in the space $W^{1,p(x)}(\Omega)$, we shall prove the Poincar\'{e}-Wirtinger's inequality in a subspace of $W^{1,p(x)}(\Omega)$.