# Newtonian Potential

— 1. Definition of Newtonian Potential —

Definition 1 Let ${\Omega}$ be a bounded open set, we define the Newtonian potential of ${f}$ is the function ${N_f}$ on ${{\mathbb R}^n}$ by

$\displaystyle N_f(x) = \int_\Omega \Gamma(x-y) f(y)\, dy,$

where ${\Gamma}$ is the fundamental solution of Laplace’s equation.

Theorem On a compact Riemannian manifold without boundary ${(M,g)}$, the equation
$\displaystyle -\Delta_g u + a(x)u = -u^p$
with ${u>0, a(x) \in \mathcal{C}^\infty(M)}$ and ${a(x)<0}$ on ${M}$, ${1 has a solution ${u \in \mathcal{C}^\infty(M)}$.