Electron. J. Diff. Equ., Vol. 2009(2009), No. 131, pp. 1-16.

Multiple solutions for nonlinear elliptic equations on Riemannian manifolds

Wenjing Chen, Jianfu Yang

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 9, 2009.
Math Subject Classifications: 35J20, 35J61, 58J05.
Key Words: Multiple Solutions; Semilinear elliptic equation; Riemannian manifold; Ljusternik-Schnirelmann category.

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Wenjing Chen
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi, 330022, China
email: wjchen1102@yahoo.com.cn
Jianfu Yang
Department of Mathematics, Jiangxi Normal University
Nanchang, Jiangxi, 330022, China
email: jfyang_2000@yahoo.com

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