Second-Order Elliptic Equation with Singularities

Hichem Boughazi

doi:10.1155/2020/4589864

Abstract

On the compact Riemannian manifold of dimension n≥5, we study the existence and regularity of nontrivial solutions for nonlinear second-order elliptic equation with singularities. At the end, we give a geometric application of the above singular equation.

Highlights

Let (M, g) be an (n ≥ 3)-dimensional compact Riemannian manifold, and let a ∈ Lp(M), where p > n/2, and f be a positive C∞(M) function on M
Ere are many results for second-order elliptic equations, but most of them are focused on bounded domains Ω of Rn or on compact Riemannian manifold (M, g), see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] for a survey
If the singular term a is replaced by (n − 2)/4(n − 1)Sg, where Sg is the scalar curvature and f 1, equation (1) becomes the famous prescribed constant scalar curvature equation which is very known in the literature as the Yamabe problem

Summary

Introduction

Let (M, g) be an (n ≥ 3)-dimensional compact Riemannian manifold, and let a ∈ Lp(M), where p > n/2, and f be a positive C∞(M) function on M. Equation (1) is one of the nonlinear second-order equations involving the singular term a and with critical Sobolev growth Such problem arises from various fields of geometry and physics. If the singular term a is replaced by (n − 2)/4(n − 1)Sg, where Sg is the scalar curvature and f 1, equation (1) becomes the famous prescribed constant scalar curvature equation which is very known in the literature as the Yamabe problem. To solve this problem, Yamabe has used the variational method, and the main difficulty of this problem is the lack of compactness for Sobolev embedding theorem. See [1] and the references therein

Notations and Preliminaries

Existence and Regularity of the Solution

Application