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 $(M^n,g)$, $n\geq 2$, $$ \Delta_p u+a(x)u^{p-1}=\lambda f(u,x),\qquad\qquad (1) $$ where $\Delta_p$ is the $p$-Laplacian, with $1<p<n$. Equation (1) generalizes an equation considered by Aubin [2], where he has considered a compact Riemannian manifold $(M,g)$, the differential equation $(p=2)$ $$ \Delta u+a(x)u=\lambda f(u,x),\qquad\qquad\qquad\,\, (2) $$ where $a(x)$ is a $C^{\infty}$ function defined on $M$, and $f(u,x)$ is a $C^{\infty}$ function defined on $\mathbb{R}\times M$. We show that equation (1) has solution $(\lambda,u)$, where $\lambda\in\mathbb{R}$, $u\geq 0$, $u\not\equiv 0$ is a function $C^{1,\alpha}$, $0<\alpha<1$, if $f\in C^{\infty}$ satisfies some growth and parity conditions.