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Solutions for a Variable Exponent Neumann Boundary Value Problems with Hardy Critical Exponent

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Abstract

In this paper, we deal with the existence of solutions for the following variable exponent system Neumann boundary value problem with Hardy critical exponent and approximate Sobolev critical growth condition

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\text {div}(\left| \nabla u\right| ^{p(x)-2}\nabla u)+a(x)\left| u\right| ^{p(x)-2}u=F_{u}(x,u,v)\text {}&{} \text { in }\Omega , \\ -\text {div}(\left| \nabla v\right| ^{q(x)-2}\nabla v)+b(x)\left| v\right| ^{q(x)-2}v=F_{v}(x,u,v)\text {}&{} \text { in }\Omega , \\ \frac{\partial u}{\partial \gamma }=0=\frac{\partial v}{\partial \gamma } \text {}\left. {}\right. &{}\text { on }\partial \Omega . \end{array} \right. \end{aligned}$$

We give several sufficient conditions for the existence of solutions, when \( F(x,\cdot ,\cdot )\) satisfies sub-(\(p(x),q(x)\)) growth condition, or super-(\( p(x),q(x)\)) growth condition and approximate Sobolev critical growth condition. Especially, we obtain the existence of infinitely many solutions, when \(F(x,\cdot ,v)\) satisfies sub-\(p(x)\) growth condition, and \(F(x,u,\cdot )\) satisfies super-\(q(x)\) growth condition.

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Acknowledgments

Foundation item: Partly supported by the key projects of Science and Technology Research of the Henan Education Department (14A110011) and the National Natural Science Foundation of China (11326161 and 10971087).

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  1. College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, 450002, Henan, China

    Guizhen Zhi & Qihu Zhang

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  1. Guizhen Zhi
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  2. Qihu Zhang
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Correspondence to Qihu Zhang.

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Dedicated to Professor Xianling Fan on his 70th birthday.

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Zhi, G., Zhang, Q. Solutions for a Variable Exponent Neumann Boundary Value Problems with Hardy Critical Exponent. Bull. Malays. Math. Sci. Soc. 38, 571–603 (2015). https://doi.org/10.1007/s40840-014-0037-5

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  • DOI: https://doi.org/10.1007/s40840-014-0037-5

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