Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 131, pp. 1-16.

Title: Multiple solutions for nonlinear elliptic equations on
       Riemannian manifolds

Authors: Wenjing Chen (Jiangxi Normal Univ., Nanchang, China)
         Jianfu Yang  (Jiangxi Normal Univ., Nanchang, China)

Abstract:
 Let $(\mathcal{M}, g)$ be a compact, connected, orientable,
 Riemannian $n$-manifold of class $C^{\infty}$
 with Riemannian metric $g$ $(n\geq 3)$. We study the
 existence of solutions to the  equation
 $$
 -\varepsilon^2\Delta_{g} u+V(x)u=K(x)|u|^{p-2}u
 $$
 on this Riemannian manifold. Here $2<p<2^{*}=2n/(n-2)$, $V(x)$
 and $K(x)$ are continuous functions. We show that the shape
 of $V(x)$ and $K(x)$ affects the  number of solutions,
 and then prove the existence of multiple solutions.

Submitted September 15, 2009. Published October 09, 2009.
Math Subject Classifications: 35J20, 35J61, 58J05.
Key Words: Multiple Solutions; Semilinear elliptic equation;
           Riemannian manifold; Ljusternik-Schnirelmann category.