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