Let $M:=S^2$ be two-dimensional the unit sphere. For a given source function $f \in H^{-1}(M) $ I want to find a weak solution $u \in H^1(M) $ that is not classical such that for all $v \in H^1(M)$ we have $$\int_M \nabla u \cdot \nabla v \, dx= \int_M fv \, dx.$$
Is there any literature where I can find examples for such problems?