Minimal support results for Schrödinger equations

2013/02/18 by Laura De Carli, De Carli, Laura, Julian Edward +5
Mathematics · #35P15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35B05 #Secondary 26D20 #math.AP #msc:26D20 #msc:35B05 #msc:35P15

paper · pdf · doi:10.48550/arxiv.1302.4335

arxiv created 2013/02/18 · arxiv updated 2013/02/19

Abstract

We consider a number of linear and non-linear boundary value problems involving generalized Schrödinger equations. The model case is -Δu=Vu for u∈ W₀^1,2(D) with D a bounded domain in \bf Rⁿ. We use the Sobolev embedding theorem, and in some cases the Moser-Trudinger inequality and the Hardy-Sobolev inequality, to 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.

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