Abstract and Applied Analysis

Existence of Solutions for the p ( x ) -Laplacian Problem with the Critical Sobolev-Hardy Exponent

Yu Mei, Fu Yongqiang, and Li Wang

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Abstract

This paper deals with the p ( x ) -Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness in W 0 1 , 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 ) - 2 u = ( h ( x ) | u | p s * ( x ) - 2 u / | x | s ( x ) ) + f ( x , u ) , x Ω , u = 0 , x Ω , where Ω N is a bounded domain, 0 Ω , 1 < p - p ( x ) p + < N , and f ( x , u ) satisfies p ( x ) -growth conditions.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 894925, 17 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495827

Digital Object Identifier
doi:10.1155/2012/894925

Mathematical Reviews number (MathSciNet)
MR2965446

Zentralblatt MATH identifier
1250.35107

Citation

Mei, Yu; Yongqiang, Fu; Wang, Li. Existence of Solutions for the $p(x)$ -Laplacian Problem with the Critical Sobolev-Hardy Exponent. Abstr. Appl. Anal. 2012 (2012), Article ID 894925, 17 pages. doi:10.1155/2012/894925. https://projecteuclid.org/euclid.aaa/1355495827


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