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We introduce the Local Increasing Regularity Method (LIRM) which allows us toget from \emph{local} a priori estimates, on solutions $u$ of a linear equation$\displaystyle Du=\omega ,$ \emph{global} ones.  As an application we shall prove that if $D$ is an elliptic lineardifferential operator of order $m$ with ${\mathcal{C}}^{\infty }$ coefficientsoperating on the sections of a complex vector bundle $\displaystyle G:=(H,\pi,M)$ over a compact Riemannian manifold $M$ without boundary and $\omega \inL^{r}_{G}(M)\cap (\mathrm{k}\mathrm{e}\mathrm{r}D^{*})^{\perp },$ then there isa $u\in W^{m,r}_{G}(M)$ such that $Du=\omega $ on $M.$ \quad Next weinvestigate the case of a compact manifold with boundary by use of the"riemannian double manifold". In the last sections we study the more delicatecase of a complete but non compact Riemannian manifold by use of adaptedweights.

The LIR Method. $L^{r}$ solutions of elliptic equation in a complete riemannian manifold