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On a supercritical Hardy-Sobolev type inequality with logarithmic term and related extremal problem

José Francisco de Oliveira, Jeferson Silva

Abstract

Our main goal is to investigate supercritical Hardy-Sobolev type inequalities with a logarithmic term and their corresponding variational problem. We prove the existence of extremal functions for the associated variational problem, despite the loss of compactness. As an application, we show the existence of weak solution to a general class of related elliptic partial differential equations with a logarithmic term.

On a supercritical Hardy-Sobolev type inequality with logarithmic term and related extremal problem

Abstract

Our main goal is to investigate supercritical Hardy-Sobolev type inequalities with a logarithmic term and their corresponding variational problem. We prove the existence of extremal functions for the associated variational problem, despite the loss of compactness. As an application, we show the existence of weak solution to a general class of related elliptic partial differential equations with a logarithmic term.
Paper Structure (16 sections, 15 theorems, 190 equations)

This paper contains 16 sections, 15 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Supercritical inequality: Proof of Theorem \ref{['C3-T1']}
  3. Sharp estimates: Proof Theorem \ref{['T2']}
  4. Case 1
  5. Case 2
  6. Case A
  7. Sub-case
  8. Sub-case
  9. Case B
  10. $(i)$
  11. $(ii)$
  12. $(iii)$
  13. Attainability: Proof of Theorem \ref{['C3-T3']}
  14. Step 1:
  15. Step 2:
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose that $\varphi\in C[0,1)$ satisfies $(h_1)$-$(h_3)$. Then $\mathcal{S}_{\theta,\tau}(\varphi)<\infty$ for any $\tau>0$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof : Proof of the Theorem \ref{['C3-T1']}
  • Lemma 2.2
  • proof
  • proof : Proof of the Corollary \ref{['C3-C1']}
  • ...and 19 more