Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence and large radial solutions for an elliptic system under finite new Keller-Osserman integral conditions

Dragos-Patru Covei

TL;DR

This work extends Keller–Osserman theory to a coupled semilinear elliptic system with radial weights by developing KO-type transforms and subharmonic barriers under finite reciprocal integral conditions. It proves the existence of infinitely many entire positive radial solutions for a nonempty set of central values, and shows this set is closed; boundary central values yield large solutions that blow up at infinity. The paper also analyzes the compatibility and distinctions between the ELRS framework and Covei2017 results, providing a common example and clarifying when each regime applies. Overall, it advances understanding of existence and asymptotic behavior for weighted, nonlinear elliptic systems with general nonlinearities.

Abstract

We study the semilinear elliptic system \[ Δu = p(|x|)\,g(v), \qquad Δv = q(|x|)\,f(u), \qquad x \in \mathbb{R}^n,\; n \geq 3, \] under new Keller--Osserman-type integral conditions on the nonlinearities $f,g$ and decay constraints on the radial weights $p,q$. Within this framework we prove: (i) existence of infinitely many entire positive radial solutions for admissible central values; (ii) closedness of the set of all admissible central values; and (iii) largeness (blow-up at infinity) of solutions at boundary points. The analysis combines comparison principles, compactness arguments, and Keller--Osserman transforms, thereby extending classical theory to coupled elliptic systems with general nonlinearities and weights.

Existence and large radial solutions for an elliptic system under finite new Keller-Osserman integral conditions

TL;DR

This work extends Keller–Osserman theory to a coupled semilinear elliptic system with radial weights by developing KO-type transforms and subharmonic barriers under finite reciprocal integral conditions. It proves the existence of infinitely many entire positive radial solutions for a nonempty set of central values, and shows this set is closed; boundary central values yield large solutions that blow up at infinity. The paper also analyzes the compatibility and distinctions between the ELRS framework and Covei2017 results, providing a common example and clarifying when each regime applies. Overall, it advances understanding of existence and asymptotic behavior for weighted, nonlinear elliptic systems with general nonlinearities.

Abstract

We study the semilinear elliptic system under new Keller--Osserman-type integral conditions on the nonlinearities and decay constraints on the radial weights . Within this framework we prove: (i) existence of infinitely many entire positive radial solutions for admissible central values; (ii) closedness of the set of all admissible central values; and (iii) largeness (blow-up at infinity) of solutions at boundary points. The analysis combines comparison principles, compactness arguments, and Keller--Osserman transforms, thereby extending classical theory to coupled elliptic systems with general nonlinearities and weights.

Paper Structure

This paper contains 14 sections, 7 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: radial formulation, local existence and Keller--Osserman transform
  3. Existence of entire positive solutions
  4. The set of central values: Nonemptiness and closedness
  5. Largeness on the edge
  6. Compatibility between Theorem \ref{['thm:large']} and Theorem 2.4 of Covei2017
  7. General framework
  8. A common particular example
  9. Applicability of the two results
  10. For ELRS:
  11. For T2.4:
  12. Why there is no contradiction
  13. Concluding remarks
  14. Appendix

Key Result

Theorem 1.2

For every $(a,b)$ in a suitable nonempty set of admissible central values $\mathcal{T}\subset (0,\infty )^{2}$, there exists an entire positive radial solution $(u,v)\in C^{2}([0,\infty ))^{2}$ of eq:system with $u(0)=a$, $v(0)=b$. Moreover, $u^{\prime }(r)\geq 0$ and $v^{\prime }(r)\geq 0$ for all

Theorems & Definitions (17)

  • Remark 1.1: Relation to Lair's result
  • Theorem 1.2: Existence
  • Theorem 1.3: Nonemptiness and closedness of $S$
  • Theorem 1.4: Largeness on the edge
  • Lemma 2.1: Local existence PetersonWood2011
  • proof
  • Lemma 2.2: Alternative PetersonWood2011
  • proof
  • Lemma 2.3: Forcing inequality
  • proof
  • ...and 7 more