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Quasilinear Lane-Emden type systems with sub-natural growth terms

Estevan Luiz da Silva, João Marcos do Ó

Abstract

Global pointwise estimates are obtained for quasilinear Lane-Emden-type systems involving measures in the "sublinear growth" rate. We give necessary and sufficient conditions for existence expressed in terms of Wolff's potential. Our approach is based on recent advances due to Kilpeläinen and Malý in the potential theory. This method enables us to treat several problems, such as equations involving general quasilinear operators and fractional Laplacian.

Quasilinear Lane-Emden type systems with sub-natural growth terms

Abstract

Global pointwise estimates are obtained for quasilinear Lane-Emden-type systems involving measures in the "sublinear growth" rate. We give necessary and sufficient conditions for existence expressed in terms of Wolff's potential. Our approach is based on recent advances due to Kilpeläinen and Malý in the potential theory. This method enables us to treat several problems, such as equations involving general quasilinear operators and fractional Laplacian.
Paper Structure (16 sections, 14 theorems, 157 equations)

This paper contains 16 sections, 14 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Systems of Wolff potentials
  3. Main results
  4. Application
  5. Related results
  6. Organization of the paper
  7. Notations and definitions
  8. Preliminaries
  9. Systems of Wolff potentials
  10. Proof of Theorem \ref{['existencewollfsystem']}
  11. Proof of Theorem \ref{['existencewolffsystemlocweaker']}
  12. Applications
  13. Proof of Theorem \ref{['solutionplaplacian general']}
  14. Proof of Theorem \ref{['solutionplaplacian general weaker']}
  15. Proof of Theorem \ref{['thm frac system']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $1<p<\infty$, $0<q_i<p-1$, $i=1,2, \; 0<\alpha<{n}/{p}$ and $\sigma\in {M}^+(\mathds{R}^n)$ satisfying general wolff finite and sigma abscont alfa p capacidade. Then there exists a solution $(u,v)$ to Syst. sistemawolff such that where $c=c(n,p,q_1,q_2,\alpha,C_{\sigma})>0$. Furthermore, $u,v \in L_{\mathrm{loc}}^{s}(\mathds{R}^n,\, \mathrm{d} \sigma)$, for every $s>0$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 23 more