Minimal support results for Schr\"odinger equations

TL;DR

This paper establishes minimal conditions involving Sobolev inequalities for the existence of solutions to linear and nonlinear Schrödinger equations, providing sharp bounds and necessary criteria.

Contribution

It derives necessary conditions for solutions to Schrödinger equations using Sobolev, Moser-Trudinger, and Hardy-Sobolev inequalities, with optimal bounds.

Findings

Necessary bounds involving potential norm and domain measure

Use of Sobolev and related inequalities for existence criteria

Results are sharp and optimal in most cases

Abstract

We consider a number of linear and non-linear boundary value problems involving generalized Schr\"odinger equations. The model case is $-\Delta u=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R^n}$. 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.