带Neumann 边界条件的$p(x)$-Kirchhoff 型椭圆系统的研究

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

中文关键词:  变分法  椭圆系统  非局部  Neumann 边界条件.

英文关键词:variational method  elliptic systems  nonlocal  Neumann boundary.

中文摘要:

      本文研究一类带Neumann 边界条件的~$p(x)$-Kirchhoff 型系统解的存在性. 借助于Ekeland变分原理和变指数Sobolev空间理论, 我们给出使得该问题存在解的合适条件.由于Poincar\'{e} 不等式在$W^{1,p(x)}(\Omega)$ 中不再成立, 我们将在$W^{1,p(x)}(\Omega)$ 的某个子空间中证明Poincar\'{e}-Wirtinger 不等式.

英文摘要:

      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 $W^{1,p(x)}(\Omega)$.