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Minimal potential results for Schrödinger equations with Neumann boundary conditions

Minimal potential results for Schrödinger equations with Neumann boundary conditions AbstractWe consider the boundary valueproblem -Δp⁢u=V⁢|u|p-2⁢u-C{-\Delta_{p}u=V|u|^{p-2}u-C}, where u∈W1,p⁢(D){u\in W^{1,p}(D)}is assumed to satisfy Neumann boundary conditions, and D is a bounded domain in ℝn{{\mathbb{R}^{n}}}.We derive necessary conditions for the existence of nontrivial solutions.These conditions usually involve a lower bound for the product of a sharp Sobolev constant and an Lp{L^{p}}norm of V.When p=n{p=n}, Orlicz norms are used.In many cases, these inequalities are best possible.Applications to linear and non-linear eigenvalue problems are also discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Minimal potential results for Schrödinger equations with Neumann boundary conditions

Edward, Julian; Hudson, Steve; Leckband, Mark
Forum Mathematicum , Volume 29 (6): 12 – Nov 1, 2017
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Publisher
de Gruyter
Copyright
© 2017 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2015-0082
Publisher site
See Article on Publisher Site

Abstract

AbstractWe consider the boundary valueproblem -Δp⁢u=V⁢|u|p-2⁢u-C{-\Delta_{p}u=V|u|^{p-2}u-C}, where u∈W1,p⁢(D){u\in W^{1,p}(D)}is assumed to satisfy Neumann boundary conditions, and D is a bounded domain in ℝn{{\mathbb{R}^{n}}}.We derive necessary conditions for the existence of nontrivial solutions.These conditions usually involve a lower bound for the product of a sharp Sobolev constant and an Lp{L^{p}}norm of V.When p=n{p=n}, Orlicz norms are used.In many cases, these inequalities are best possible.Applications to linear and non-linear eigenvalue problems are also discussed.

Journal

Forum Mathematicum – de Gruyter

Published: Nov 1, 2017

References

  • Bounds on the eigenvalues of the Laplace and Schrödinger operators
  • Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition
  • Quasiregular mappings and the p-Laplace operator