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arXiv · 1302.4335

Minimal support results for Schr\"odinger equations

Laura De Carli·Julian Edward·Steve Hudson·Mark Leckband
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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.

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Laura De Carli, Julian Edward, Steve Hudson, Mark Leckband. 2013-02-18. Minimal support results for Schr\"odinger equations. https://arxiv.org/abs/1302.4335

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