The Allen-Cahn equation on the complete Riemannian manifolds of finite volume

The semi-linear, elliptic PDE AC(epsilon)(u):= -epsilon(2)Delta u+W '(u)=0 is called the Allen-Cahn equation. In this article we will prove the existence of finite energy solution to the Allen-Cahn equation on certain complete, non-compact manifolds. More precisely, suppose Mn+1 (with n+1 >= 3) is a complete Riemannian manifold of finite volume. Then there exists epsilon(0)>0, depending on the ambient Riemannian metric, such that for all 0 < epsilon <= epsilon(0), there exists u(epsilon ): M ->(-1,1) satisfying AC(epsilon)(u(epsilon)) = 0 with the energy E-epsilon(u(epsilon))<infinity and the Morse index Ind(u(epsilon))<= 1. Moreover, 0<lim inf(epsilon -> 0)E(epsilon)(u(epsilon)) <= lim sup(epsilon -> 0)E(epsilon)(u(epsilon)) < infinity. Our result is motivated by the theorem of Chambers-Liokumovich and Song, which says that M contains a complete minimal hypersurface Sigma with 0<H-n(Sigma)<infinity. This theorem can be recovered from our result. (c) 2024 Elsevier Inc. All rights reserved.