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Minimizers and best constants for a weighted critical Sobolev inequality involving the polyharmonic operator

José Francisco de Oliveira, Jeferson Silva

Abstract

Our main goal is to explicitly compute the best constant for the Sobolev-type inequality involving the polyharmonic operator obtained in (Analysis and Applications 22, pp. 1417-1446, 2024). To achieve this goal, we also establish both regularity and classification results for a generalized critical polyharmonic equation in the radial setting.

Minimizers and best constants for a weighted critical Sobolev inequality involving the polyharmonic operator

Abstract

Our main goal is to explicitly compute the best constant for the Sobolev-type inequality involving the polyharmonic operator obtained in (Analysis and Applications 22, pp. 1417-1446, 2024). To achieve this goal, we also establish both regularity and classification results for a generalized critical polyharmonic equation in the radial setting.
Paper Structure (15 sections, 16 theorems, 199 equations, 1 table)

This paper contains 15 sections, 16 theorems, 199 equations, 1 table.

Table of Contents

  1. Introduction and main results
  2. Sobolev spaces with weights
  3. Main results
  4. Regularity result and Minimizers behavior
  5. Proof Theorem \ref{['Teorema unicidade']}
  6. Step 1:
  7. Step 2:
  8. Proof of Step $1$
  9. Case 1:
  10. Case 2:
  11. Case 3:
  12. Case 4:
  13. Proof of step 2
  14. Proof Corollary \ref{['cor 1.2']}
  15. Proof Theorem \ref{['Theorem 1.3']}

Key Result

Theorem 1.1

Suppose $\alpha-2m+1>0$ and let $u \in \mathcal{D}^{m,2}_{\infty}(\alpha)$ be weak solution problema m, that is, Then, $u\in C^{2m}(0,\infty)$ and solves the equation problema m. In addition, $u$ satisfies boundary conditions and and the identity

Theorems & Definitions (32)

  • Theorem 1.1: Regularity
  • Theorem 1.2: classification
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Conjecture 1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 22 more