Existence and nonexistence of energy solutions for linear elliptic equations involving Hardy-type potentials

Konstantinos T. Gkikas, Existence and nonexistence of energy solutions for linear elliptic equations involving Hardy-type potentials, Indiana Univ. Math. J. 58 (2009), 2317-2346

Abstract

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Let $Ω \subset R^{n}$ be an open domain that contains the origin.
We find conditions on the potential $V$ which ensure the nonexistence of positive $H^{1} (Ω)$ solutions for linear elliptic problems with Hardy-type potentials. For instance, we prove the nonexistence of nontrivial solutions in $H^{1} (Ω)$ for the equation $- Δ u = \frac{{(n - 2)}^{2}}{4} \frac{u}{{| x |}^{2}} + b V u .$ The results depend on an integral assumption on the potential $V$ (see (1)).
We also give an example establishing that this integral assumption on $V$ is optimal.

Added by the Editors: In order to make the Abstract complete, here is the equation to which the abstract alludes, and which appears later in the paper:

\begin{matrix} \int_{Ω} | V |^{n / 2} X_{1}^{1 - n} d x < \infty, (1) \end{matrix}

Department of Mathematics
University of Crete
71409 Heraklion, Greece
E-mail address : kugkikas@math.uoc.gr

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