Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on borderline Brezis-Nirenberg type problems

Julian Haddad, Marcos Montenegro

TL;DR

This work analyzes borderline Brezis–Nirenberg-type problems for divergence-form elliptic operators concentrating ellipticity at a boundary point. It introduces interior $\alpha$-singular boundary points and a constrained, variational framework with $\Phi_A$ and $\Psi_{a,p}$ on unit Sobolev-type manifolds, using bubbles to obtain strict energy inequalities relative to the Sobolev constants $K(n,2)$ and $K(n,p)$. By constructing boundary-centered bubbles and performing precise scaling estimates, it proves the existence of positive $C^1$ solutions for all $\lambda$ below the first eigenvalue, under specified growth exponents $\gamma$, $\sigma$ and boundary singularity conditions $\alpha$. These results extend the classical Brezis–Nirenberg theory to non-smooth domains with boundary ellipticity concentration and clarify how boundary geometry governs existence thresholds in border-case regimes.

Abstract

We study the effect of ellipticity points on the boundary in the Brezis-Nirenberg problem for elliptic operators in divergence form.

A note on borderline Brezis-Nirenberg type problems

TL;DR

This work analyzes borderline Brezis–Nirenberg-type problems for divergence-form elliptic operators concentrating ellipticity at a boundary point. It introduces interior -singular boundary points and a constrained, variational framework with and on unit Sobolev-type manifolds, using bubbles to obtain strict energy inequalities relative to the Sobolev constants and . By constructing boundary-centered bubbles and performing precise scaling estimates, it proves the existence of positive solutions for all below the first eigenvalue, under specified growth exponents , and boundary singularity conditions . These results extend the classical Brezis–Nirenberg theory to non-smooth domains with boundary ellipticity concentration and clarify how boundary geometry governs existence thresholds in border-case regimes.

Abstract

We study the effect of ellipticity points on the boundary in the Brezis-Nirenberg problem for elliptic operators in divergence form.

Paper Structure

This paper contains 3 sections, 2 theorems, 34 equations.

Table of Contents

  1. Introduction and statements
  2. Bubbles estimates
  3. Proof of Theorems \ref{['T.1']} and \ref{['T.2']}

Key Result

Theorem 1

Let $\Omega$ be a bounded open subset of $\Bbb{R}^n$ with $n \geq 5$. Assume that the determinant of $A(x)$ attains its global minimum at a point $x_0$ on the boundary of $\Omega$ such that the comparison (H1) is satisfied with $\gamma > \frac{2n - 4}{n - 4}$. Assume also that the boundary of $\Omeg

Theorems & Definitions (3)

  • Definition 1
  • Theorem 1
  • Theorem 2