On Nonlocal Elliptic Systems of $p(x)$-Kirchhoff-Type under Neumann Boundary Condition
Received:February 08, 2012  Revised:September 04, 2012
Key Word: variational method   elliptic systems   nonlocal   Neumann boundary.  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11261052).
Author NameAffiliation
Guowei DAI Department of Mathematics, Northwest Normal University, Gansu 730070, P. R. China 
Xiaoyan LI Department of Mathematics, Northwest Normal University, Gansu 730070, P. R. China 
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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 $W^{1,p(x)}(\Omega)$.
Citation:
DOI:10.3770/j.issn:2095-2651.2013.04.007
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