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Existence of Solutions for the ð ‘ ( ð ‘¥ ) -Laplacian Problem with the Critical Sobolev-Hardy Exponent

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  • Yu Mei
  • Fu Yongqiang
  • Li Wang

Abstract

This paper deals with the ð ‘ ( ð ‘¥ ) -Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness in ð ‘Š 0 1 , ð ‘ ( ð ‘¥ ) ( Ω ) space is established, then by applying it we obtain the existence of solutions for the following ð ‘ ( ð ‘¥ ) -Laplacian problem: − d i v ( | ∇ ð ‘¢ | ð ‘ ( ð ‘¥ ) − 2 ∇ ð ‘¢ ) + | ð ‘¢ | ð ‘ ( ð ‘¥ ) − 2 ð ‘¢ = ( â„Ž ( ð ‘¥ ) | ð ‘¢ | ð ‘ âˆ— ð ‘ ( ð ‘¥ ) − 2 ð ‘¢ / | ð ‘¥ | ð ‘ ( ð ‘¥ ) ) + ð ‘“ ( ð ‘¥ , ð ‘¢ ) , ð ‘¥ ∈ Ω , ð ‘¢ = 0 , ð ‘¥ ∈ 𠜕 Ω , where Ω ⊂ â„ ð ‘ is a bounded domain, 0 ∈ Ω , 1 < ð ‘ âˆ’ ≤ ð ‘ ( ð ‘¥ ) ≤ ð ‘ + < ð ‘ , and ð ‘“ ( ð ‘¥ , ð ‘¢ ) satisfies ð ‘ ( ð ‘¥ ) -growth conditions.

Suggested Citation

  • Yu Mei & Fu Yongqiang & Li Wang, 2012. "Existence of Solutions for the ð ‘ ( ð ‘¥ ) -Laplacian Problem with the Critical Sobolev-Hardy Exponent," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-17, August.
    • Handle: RePEc:hin:jnlaaa:894925
    DOI: 10.1155/2012/894925
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