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Optimal result involving the Green's function

Zakaria Boucheche

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.

Optimal result involving the Green's function

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.
Paper Structure (2 sections, 3 theorems, 44 equations)

This paper contains 2 sections, 3 theorems, 44 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['optimal']}

Key Result

Theorem 1.1

Let $0< \mu< \pi^2.$ Assume that $K(x)$ satisfies the assumptions $\mathbf{(K)}$ and $\mathbf{(K_{\eta}^1)}.$ Then there exists a constant $\bar{\eta}> 0$ depending on $f_1(t)$ and $K(o)$ such that if $\,\,0\leq \eta\leq \bar{\eta},$ then the problem $\mathbf{(BN)}_{\eta}^1$ admits a solution if and

Theorems & Definitions (3)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2