Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the equation $$ \Delta_g u = 4 \pi \delta_a -1 $$ has a solution $u \in W^{1,p}$, $p \in (1,2)$.
This is an attempt I made: the weak formulation of this equation is $$ -\int_{S^2} (\nabla \phi, \nabla u)_g d\mu_g = 4 \pi\phi(a) -\int_{S^2} \phi d\mu_g , $$ where integration is performed with respect to the measure induced by $g$, and $(\cdot, \cdot)_g$ is the inner product w.r.t. $g$. We notice that the RHS of the previous equation is an element of $(W^{1,p})'$, for $p > 2$, by Sobolev embedding. I would then like to use some kind of representation of $(W^{1,p})'$ to conclude that this functional can be represented as $$ \int_{S^2} (\nabla \phi, \nabla u)_g d\mu_g $$ for some $u$. Are there any ways to conclude? I guess this is pretty standard stuff, I would be very grateful if anyone could provide me a reference.