Remarks on eigenvalue problems involving the p(x)-Laplacian

By Xianling Fan. 2009

68 citations recorded by the source corpus.

Abstract This paper deals with the eigenvalue problem involving the p ( x ) -Laplacian of the form { − div ( | ∇ u | p ( x ) − 2 ∇ u ) = λ | u | q ( x ) − 2 u in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R N , p ∈ C 0 ( Ω ¯ ) , inf x ∈ Ω p ( x ) > 1 , q ∈ L ∞ ( Ω ) , 1 ⩽ q ( x ) ⩽ q ( x ) + e p * ( x ) for x ∈ Ω , e is a positive constant, p * ( x ) is the Sobolev critical exponent. It is shown that for every t > 0 , the problem has at least one sequence of solutions { ( u n , t , λ n , t ) } such that ∫ Ω 1 p ( x ) | ∇ u n , t | p ( x ) = t and λ n , t → ∞ as n → ∞ . The principal eigenvalues for the problem in several important cases are discussed especially. The similarities and the differences in the eigenvalue problem between the variable exponent case and the constant exponent case are exposed. Some known results on the eigenvalue problem are extended.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b8fc8cf0b1a62c530091207b3b657692baba5924

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In Applied Mathematics

  1. P. Hasto
  2. Petteri Harjulehto
  3. Dušan D. Repovš
  4. Matti Nuortio
  5. Ut V. Le
  6. V. Ruadulescu
  7. Mihai Mihuailescu
  8. Vicentiu D. Rădulescu

In Computational Theory and Mathematics

  1. P. Hasto
  2. Petteri Harjulehto
  3. Dušan D. Repovš
  4. V. Ruadulescu
  5. Mihai Mihuailescu
  6. Matti Nuortio
  7. Ut V. Le
  8. Vicentiu D. Rădulescu

In Mathematical Physics

  1. P. Hasto
  2. Petteri Harjulehto
  3. Dušan D. Repovš
  4. Mihai Mihuailescu
  5. Matti Nuortio
  6. Ut V. Le
  7. V. Ruadulescu
  8. Jan Vyb'iral

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