On the study of a class of non-linear differential equations on compact Riemannian Manifolds

We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds $(M^n,g),n\ge 2$, \Delta_p u + a(x)u^{p-1} = \lambda f(u,x), (E2) where ${\mathrm{\Delta }}_p$ is the $p-$laplacian, with $1<p<n$. The equation (E2) generalizes a equation considered by Aubin, where he has considered, a compact Riemannian manifold $(M,g)$, the differential equation ($p=2$) \Delta u + a(x)u = \lambda f(u,x), (E1) where $a(x)$ is a $C^{\mathrm{\infty }}$ function defined on $M$ and $f(u,x)$ is a $C^{\mathrm{\infty }}$ function defined on $\mathbb{R}\times M$. We show that the equation (E2) has solution $(\lambda ,u)$, where $\lambda \in \mathbb{R}$, $u\ge 0$, $u\equiv ̸0$ is a function $C^{1,\alpha }$, $0<\alpha <1$, if $f\in C^{\mathrm{\infty }}$ satisfies some growth and parity conditions.