Existence and regularity of positive solutions to elliptic equations of Schr\"{o}dinger type

TL;DR

This paper establishes the existence and regularity of positive solutions to Schrödinger-type elliptic equations with distributional potentials, linking form boundedness conditions to solution existence and positivity.

Contribution

It provides a novel characterization connecting form boundedness of potentials with the existence of positive solutions in elliptic equations of Schrödinger type.

Findings

Equivalence between form boundedness and positive solution existence.

Necessary and sufficient conditions for form boundedness with sharp bounds.

Examples illustrating the relationship between form boundedness and positivity.

Abstract

We prove the existence of positive solutions with optimal local regularity to homogeneous elliptic equations of Schr\"{o}dinger type, under only a form boundedness assumption on $\sigma \in D'(\Omega)$ and ellipticity assumption on $\mathcal{A}\in L^\infty(\Omega)^{n\times n}$, for an arbitrary open set $\Omega\subseteq \mathbf{R}^n$. We demonstrate that there is a two way correspondence between the form boundedness and the existence of positive solutions to this equation, as well as weak solutions to certain elliptic equations with quadratic nonlinearity in the gradient. As a consequence, we obtain necessary and sufficient conditions for both the form-boundedness (with a sharp upper form bound) and the positivity of the quadratic form of the Schr\"{o}dinger type operator with arbitrary distributional potential $\sigma \in D'(\Omega)$, and give examples clarifying the relationship between these two properties.