Optimal result involving the Green's function

TL;DR

This paper explores the conditions under which positive solutions exist or do not exist for a nonlinear elliptic equation with a critical Sobolev exponent in a 3D ball, using variational methods.

Contribution

It introduces a novel approach to analyze the existence threshold for solutions involving the Green's function in critical elliptic problems.

Findings

Identifies the precise borderline between existence and non-existence of solutions.

Develops a new testing function approach for variational problems.

Provides insights into the role of Green's function in critical elliptic equations.

Abstract

We investigate a borderline between existence and non-existence of positive solution for a nonlinear elliptic equation involving a critical Sobolev exponent in three-dimensional ball. The method is relied on a suitable choice of the functions used to test two natural ingredients for the associated variational problem.