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Existence problems for the p-Laplacian

  • Julian Edward , Steve Hudson and Mark Leckband
Published/Copyright: March 27, 2013
Forum Mathematicum
From the journal Forum Mathematicum Volume 27 Issue 2

Abstract

We consider a number of boundary value problems involving the p-Laplacian. The model case is -Δpu=V|u|p-2u for u∈W01,p(D) with D a bounded domain in ℝn. We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of V, the measure of D, and a sharp Sobolev constant. In most cases, these inequalities are best possible. Applications to non-linear eigenvalue problems are also discussed.

Keywords: p-Laplacian
MSC: 35J92; 35P15

The authors wish to thank the anonymous referee whose numerous comments improved the presentation of this paper. They also thank Laura De Carli for several useful conversations.

Received: 2012-9-24
Revised: 2012-12-31
Published Online: 2013-3-27
Published in Print: 2015-3-1

© 2015 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Near abelian profinite groups
  3. On Calabi–Yau threefolds associated to a web of quadrics
  4. On the partial fraction decomposition of the restricted partition generating function
  5. Projectively flat Finsler manifolds with infinite dimensional holonomy
  6. Weighted version of Carleson measure and multilinear Fourier multiplier
  7. Injective objects and retracts of Fraïssé limits
  8. On hypersurfaces containing projective varieties
  9. Additive reducts of algebraically closed valued fields
  10. On the subconvexity problem for GL(3) × GL(2) L-functions
  11. An algebraic property of Hecke operators and two indefinite theta series
  12. Sum formula of multiple Hurwitz-zeta values
  13. Classification of homogeneous holomorphic hermitian principal bundles over G/K
  14. Integrality properties of the CM-values of certain weak Maass forms
  15. Heat kernel for fractional diffusion operators with perturbations
  16. New class of multiple weights and new weighted inequalities for multilinear operators
  17. Balanced Hermitian geometry on 6-dimensional nilmanifolds
  18. Extensions of tame algebras and finite group schemes of domestic representation type
  19. Weighted estimates for multilinear Fourier multipliers
  20. Cluster tilting vs. weak cluster tilting in Dynkin type A infinity
  21. From ring epimorphisms to universal localisations
  22. Brauer algebras of type B
  23. Existence problems for the p-Laplacian
  24. Averaging operators over nondegenerate quadratic surfaces in finite fields
  25. Lawson homology for abelian varieties
https://doi.org/10.1515/forum-2012-0142
Keywords for this article
p-Laplacian

Articles in the same Issue

  1. Frontmatter
  2. Near abelian profinite groups
  3. On Calabi–Yau threefolds associated to a web of quadrics
  4. On the partial fraction decomposition of the restricted partition generating function
  5. Projectively flat Finsler manifolds with infinite dimensional holonomy
  6. Weighted version of Carleson measure and multilinear Fourier multiplier
  7. Injective objects and retracts of Fraïssé limits
  8. On hypersurfaces containing projective varieties
  9. Additive reducts of algebraically closed valued fields
  10. On the subconvexity problem for GL(3) × GL(2) L-functions
  11. An algebraic property of Hecke operators and two indefinite theta series
  12. Sum formula of multiple Hurwitz-zeta values
  13. Classification of homogeneous holomorphic hermitian principal bundles over G/K
  14. Integrality properties of the CM-values of certain weak Maass forms
  15. Heat kernel for fractional diffusion operators with perturbations
  16. New class of multiple weights and new weighted inequalities for multilinear operators
  17. Balanced Hermitian geometry on 6-dimensional nilmanifolds
  18. Extensions of tame algebras and finite group schemes of domestic representation type
  19. Weighted estimates for multilinear Fourier multipliers
  20. Cluster tilting vs. weak cluster tilting in Dynkin type A infinity
  21. From ring epimorphisms to universal localisations
  22. Brauer algebras of type B
  23. Existence problems for the p-Laplacian
  24. Averaging operators over nondegenerate quadratic surfaces in finite fields
  25. Lawson homology for abelian varieties
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