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Minimal potential results for Schrödinger equations with Neumann boundary conditions by Julian Edward; Steve Hudson; Mark Leckband is a Mathematics article available to read on EtoBox.

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## Abstract We consider the boundary value problem - Δ p ⁢ u = V ⁢ | u | p - 2 ⁢ u - C {-\Delta\_{p}u=V|u|^{p-2}u-C} , where u ∈ W 1 , p ⁢ ( D ) {u\in W^{1,p}(D)} is assumed to satisfy Neumann boundary conditions, and __D__ is a bounded domain in R n {{\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 L p {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.

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Author

Julian Edward; Steve Hudson; Mark Leckband

Publisher

Walter de Gruyter GmbH

Published

2017

Language

EN

Field

Mathematics (Physical Sciences)