Existence of Solutions for the p(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent
This paper deals with the p(x)-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness in W01,p(x)(Ω) space is established, then by applying it we obtain the existence of solutions for the following p(x)-Laplacian problem: -div (|∇u|p(x)-2∇...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/894925 |
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| author | Yu Mei Fu Yongqiang Li Wang |
| author_facet | Yu Mei Fu Yongqiang Li Wang |
| author_sort | Yu Mei |
| collection | DOAJ |
| description | This paper deals with the p(x)-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness in W01,p(x)(Ω) space is established, then by applying it we obtain the existence of solutions for the following p(x)-Laplacian problem: -div (|∇u|p(x)-2∇u)+|u|p(x)-2u=(h(x)|u|ps*(x)-2u/|x|s(x))+f(x,u), x∈Ω, u=0, x∈∂Ω, where Ω⊂ℝN is a bounded domain, 0∈Ω, 1<p-≤p(x)≤p+<N, and f(x,u) satisfies p(x)-growth conditions. |
| format | Article |
| id | doaj-art-15c085cd5ca14b99ac82840681b85865 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-15c085cd5ca14b99ac82840681b858652025-02-03T01:21:45ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/894925894925Existence of Solutions for the p(x)-Laplacian Problem with the Critical Sobolev-Hardy ExponentYu Mei0Fu Yongqiang1Li Wang2Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, ChinaDepartment of Mathematics, Harbin Institute of Technology, Harbin 150001, ChinaDepartment of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, ChinaThis paper deals with the p(x)-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness in W01,p(x)(Ω) space is established, then by applying it we obtain the existence of solutions for the following p(x)-Laplacian problem: -div (|∇u|p(x)-2∇u)+|u|p(x)-2u=(h(x)|u|ps*(x)-2u/|x|s(x))+f(x,u), x∈Ω, u=0, x∈∂Ω, where Ω⊂ℝN is a bounded domain, 0∈Ω, 1<p-≤p(x)≤p+<N, and f(x,u) satisfies p(x)-growth conditions.http://dx.doi.org/10.1155/2012/894925 |
| spellingShingle | Yu Mei Fu Yongqiang Li Wang Existence of Solutions for the p(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent Abstract and Applied Analysis |
| title | Existence of Solutions for the p(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent |
| title_full | Existence of Solutions for the p(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent |
| title_fullStr | Existence of Solutions for the p(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent |
| title_full_unstemmed | Existence of Solutions for the p(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent |
| title_short | Existence of Solutions for the p(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent |
| title_sort | existence of solutions for the p x laplacian problem with the critical sobolev hardy exponent |
| url | http://dx.doi.org/10.1155/2012/894925 |
| work_keys_str_mv | AT yumei existenceofsolutionsforthepxlaplacianproblemwiththecriticalsobolevhardyexponent AT fuyongqiang existenceofsolutionsforthepxlaplacianproblemwiththecriticalsobolevhardyexponent AT liwang existenceofsolutionsforthepxlaplacianproblemwiththecriticalsobolevhardyexponent |