IUMJ

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

Authors: Konstantinos T. Gkikas

Issue: Volume 58 (2009), Issue 5, 2317-2346

Abstract:

Let $\Omega\subset\mathbb{R}^n$ be an open domain that contains the origin. We find conditions on the potential $V$ which ensure the nonexistence of positive $H^1(\Omega)$ solutions for linear elliptic problems with Hardy-type potentials. For instance, we prove the nonexistence of nontrivial solutions in $H^1(\Omega)$ for the equation \[-\Delta u = \frac{(n - 2)^2}{4}\frac{u}{|x|^2} + bVu.\] The results depend on an integral assumption on the potential $V$ (see (1.4)). We also give an example establishing that this integral assumption on $V$ is optimal.