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Extremal functions for a supercritical k-Hessian inequality of Sobolev-type

José Francisco de Oliveira, Pedro Ubilla

Abstract

Our main purpose in this paper is to investigate a supercritical Sobolev-type inequality for the $k$-Hessian operator acting on $Φ^{k}_{0,\mathrm{rad}}(B)$, the space of radially symmetric $k$-admissible functions on the unit ball $B\subset\mathbb{R}^{N}$. We also prove both the existence of admissible extremal functions for the associated variational problem and the solvability of a related $k$-Hessian equation with supercritical growth.

Extremal functions for a supercritical k-Hessian inequality of Sobolev-type

Abstract

Our main purpose in this paper is to investigate a supercritical Sobolev-type inequality for the -Hessian operator acting on , the space of radially symmetric -admissible functions on the unit ball . We also prove both the existence of admissible extremal functions for the associated variational problem and the solvability of a related -Hessian equation with supercritical growth.

Paper Structure

This paper contains 17 sections, 18 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. The inequality: Proof of Theorem \ref{['thm1']}
  3. Proof Theorem \ref{['thm1']}
  4. Proof Corollary \ref{['coro1']}
  5. An auxiliary extremal problem
  6. Step 1:
  7. Step 2:
  8. Step 3:
  9. Proof of Step 1
  10. Proof of Step 2
  11. Proof of Step 3
  12. Existence of $k$-admissible Extremals: Proof of Theorem \ref{['thm2']}
  13. $k$-admissible solution to the related supercritical equation
  14. (1)
  15. (2)
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\alpha>0$ be real number and assume $1\le k< N/2$. Then where $\Phi^{k}_{0,\mathrm{rad}}(B)$ is the subspace of radially symmetric functions in $\Phi^{k}_0(B)$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Definition 3.2
  • ...and 22 more