Singular elliptic equation involving the GJMS operator on the standard unit sphere

TL;DR

This paper constructs an example of a positive function satisfying certain conditions on the standard sphere, thereby improving previous results on the existence of solutions to a singular elliptic equation involving the GJMS operator.

Contribution

It provides an explicit example on the sphere that meets the conditions for multiple solutions to a GJMS-related elliptic equation, enhancing prior theoretical results.

Findings

Constructed an explicit positive function on the sphere

Improved conditions for solution existence on the sphere

Demonstrated multiple solutions for the elliptic equation

Abstract

Given a Riemannian compact manifold (M,g) of dimension n>4, we have proven in [1] under some conditions that the equation : Pg(u) = Bu +Au2+Cu (1) where Pg is the GJMS-operator, n = dim(M) > 2k, A, B and C are smooth positive functions on M, p > 1 and 2] denotes the critical Sobolev admits twodistinct positive solutions. The proof of this result is essentially based on the given smooth function ' > 0 with norm k'kPg = 1 fulfilling some conditions ( see Theorem 3 in [1]). In this note we construct an example of such function on the unit standard sphere (Sn; h). Con- sequently the conditions of the Theorem are improved in the case of (Sn; h)