Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of positive solutions for a class of almost critical problems on an annulus

Gabriele Mancini, Giuseppe Mario Rago, Giusi Vaira

TL;DR

This work analyzes the existence of multi-peak (bubble) positive solutions to slightly subcritical or slightly supercritical elliptic problems on an annulus with Dirichlet boundary conditions. The authors perform a finite-dimensional reduction around symmetric k-point configurations, exploiting the Green and Robin functions to derive a reduced energy whose leading term involves the interaction function Λ1(r). They establish a sharp threshold ρ_k for the annulus geometry: when the hole is sufficiently small or sufficiently large (depending on the ε-regime and problem type), stable critical points exist for the reduced functional, yielding k-bubble solutions, while in the opposite regime no such solutions exist. The analysis hinges on precise spectral properties of the circulant interaction matrix and intricate estimates of Gegenbauer polynomials, covering dimensions N=3,4, and N≥5. The results extend and unify previous findings, clarifying how domain geometry governs bubbling phenomena in near-critical elliptic problems on annuli.

Abstract

In this paper we will consider multi-peaks positive solutions for a class of slightly subcritical or slightly supercritical elliptic problems on an annulus with Dirichlet boundary conditions. By using the explicit form of the Green function and of the Robin function on the annulus, we prove that the annulus becomes thinner and thinner when the number of bumps increases for the slightly subcritical case, while the hole of the annulus is very small for the slightly supercritical case.

Existence of positive solutions for a class of almost critical problems on an annulus

TL;DR

This work analyzes the existence of multi-peak (bubble) positive solutions to slightly subcritical or slightly supercritical elliptic problems on an annulus with Dirichlet boundary conditions. The authors perform a finite-dimensional reduction around symmetric k-point configurations, exploiting the Green and Robin functions to derive a reduced energy whose leading term involves the interaction function Λ1(r). They establish a sharp threshold ρ_k for the annulus geometry: when the hole is sufficiently small or sufficiently large (depending on the ε-regime and problem type), stable critical points exist for the reduced functional, yielding k-bubble solutions, while in the opposite regime no such solutions exist. The analysis hinges on precise spectral properties of the circulant interaction matrix and intricate estimates of Gegenbauer polynomials, covering dimensions N=3,4, and N≥5. The results extend and unify previous findings, clarifying how domain geometry governs bubbling phenomena in near-critical elliptic problems on annuli.

Abstract

In this paper we will consider multi-peaks positive solutions for a class of slightly subcritical or slightly supercritical elliptic problems on an annulus with Dirichlet boundary conditions. By using the explicit form of the Green function and of the Robin function on the annulus, we prove that the annulus becomes thinner and thinner when the number of bumps increases for the slightly subcritical case, while the hole of the annulus is very small for the slightly supercritical case.
Paper Structure (12 sections, 11 theorems, 146 equations)

This paper contains 12 sections, 11 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The reduction scheme
  4. About the function $\Lambda(r)$
  5. Critical points of the reduced problem
  6. The sign of $\Lambda_1(r)$
  7. Positivity of $\Lambda_1(r)$
  8. The case $N=3$
  9. The case $N=4$
  10. The case $N\geq 5$
  11. Negativity of $\Lambda_1(r)$
  12. Proof of Theorems \ref{['main1']}, \ref{['main2']}

Key Result

Theorem 1.1

Let $k\geq 2$ and $N\geq 3$ be fixed. Then there exists $\rho_k\in (0, 1)$ such that:

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Theorem 2.5: GV, Theorem $2.1$
  • Definition 2.6
  • ...and 10 more