# Vanishing solution of the Poisson equation at infinity

Hi, I am interested in finding some vanish bahavior at infinity of the solutions of this kind of equations:

$-\Delta\phi+a(x)\phi=b(x)$

where $a(x), b(x)\in L^{p}$ with $1\leq p\leq 3$. Besides $\phi\in\mathcal{D}^{1,2}(\mathbb{R}^3)$ which is the completion of $C_0^{\infty}(\mathbb{R}^3)$ with respect to the norm $\|\phi\|_{\mathcal{D^{1,2}}}^2=\int|\nabla u|^2$.

Well, I think that using Potential Theory I can get this kind of property for the Poisson equation, and then mabye the same is true for my problem.

Any ideas are welcame.

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Is $a(x)$ positive? –  timur Dec 16 '11 at 23:07