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.

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A Perron-type method

I am now reviewing Perron’s method and it is a fun to go over a similar argument like this. The materials come from the lecture by Prof. Li and T.Aubin’s book.

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<p<\infty} has a solution {u \in \mathcal{C}^\infty(M)}.

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