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Entire Large Solutions for Competitive Semilinear Elliptic Systems with General Nonlinearities Satisfying Keller--Osserman Conditions

Dragos-Patru Covei

TL;DR

The paper addresses the existence of positive entire large solutions for a competitive semilinear elliptic system with general nonlinearities satisfying Keller–Osserman type growth. It adapts Lair's barrier/monotone-iteration framework to a broader nonlinear setting, reducing the system to a scalar equation for $W=u+v$ and applying a Keller–Osserman transform $H$ to obtain a quantitative blow-up mechanism via a two-step radial integration. The main contributions include a unified KO-integral criterion that covers both critical and supercritical growth, explicit lower bounds for $W$ in terms of $H^{-1}$, and compatibility with existing bounded-solution results, thereby recovering Lair's results as a special case. This work broadens the applicability of large-solution existence theory to general nonlinearities and provides a robust toolkit for future extensions to nonradial or multi-component systems.

Abstract

We generalize a theorem of Lair concerning the existence of positive entire large solutions to competitive semilinear elliptic systems. While Lair's original result \cite{Lair2025} was established for power-type nonlinearities, our work extends the theory to a broad class of general nonlinearities satisfying a Keller--Osserman-type growth condition. The proof follows the same conceptual framework monotone iteration to construct global positive solutions, reduction to a scalar inequality for the sum of the components, application of a Keller--Osserman transform, and a two-step radial integration argument but replaces the explicit power-law growth with a general monotone envelope function. This approach yields a unified and verifiable criterion for the existence of large solutions in terms of the Keller--Osserman integral, thereby encompassing both critical and supercritical growth regimes within a single analytical setting.

Entire Large Solutions for Competitive Semilinear Elliptic Systems with General Nonlinearities Satisfying Keller--Osserman Conditions

TL;DR

The paper addresses the existence of positive entire large solutions for a competitive semilinear elliptic system with general nonlinearities satisfying Keller–Osserman type growth. It adapts Lair's barrier/monotone-iteration framework to a broader nonlinear setting, reducing the system to a scalar equation for and applying a Keller–Osserman transform to obtain a quantitative blow-up mechanism via a two-step radial integration. The main contributions include a unified KO-integral criterion that covers both critical and supercritical growth, explicit lower bounds for in terms of , and compatibility with existing bounded-solution results, thereby recovering Lair's results as a special case. This work broadens the applicability of large-solution existence theory to general nonlinearities and provides a robust toolkit for future extensions to nonradial or multi-component systems.

Abstract

We generalize a theorem of Lair concerning the existence of positive entire large solutions to competitive semilinear elliptic systems. While Lair's original result \cite{Lair2025} was established for power-type nonlinearities, our work extends the theory to a broad class of general nonlinearities satisfying a Keller--Osserman-type growth condition. The proof follows the same conceptual framework monotone iteration to construct global positive solutions, reduction to a scalar inequality for the sum of the components, application of a Keller--Osserman transform, and a two-step radial integration argument but replaces the explicit power-law growth with a general monotone envelope function. This approach yields a unified and verifiable criterion for the existence of large solutions in terms of the Keller--Osserman integral, thereby encompassing both critical and supercritical growth regimes within a single analytical setting.

Paper Structure

This paper contains 7 sections, 3 theorems, 89 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof of the Theorems
  4. Proof of Theorem \ref{['lem:G-nonempty']}
  5. Proof of Theorem \ref{['lem:Gphi']}
  6. Proof of Theorem \ref{['thm:main']}
  7. Conclusion

Key Result

Theorem 2.1

Let $\alpha_{0} > 0$, $\beta_{0} > 0$. Then $\mathcal{G}$ is nonempty and has the property that if $(\alpha_{0}, \beta_{0}) \in \mathcal{G}$ and $0 < \alpha < \alpha_{0}$, $0 < \beta < \beta_{0}$, then $(\alpha, \beta) \in \mathcal{G}$.

Figures (4)

  • Figure 1: Existence region in the $(\alpha,\beta)$-plane: $\mathcal{G}$ (green), $\mathcal{G}_{\delta}$ (yellow), and $\mathcal{E}(\mathcal{G}_{\delta})$ (blue).
  • Figure 2: $W(r)\rightarrow +\infty$ forces at least one component to diverge; under the Keller--Osserman condition both diverge. Solid lines: without KO; dashed lines: with KO.
  • Figure 3: Schematic of the first contact $r_{\ast }$ where $u$ (blue) and $y$ (red) intersect: solid for $r<r_{\ast }$, dashed for $r>r_{\ast }$. On $(R_{0},r_{\ast })$ we have $w=u-y>0$ and $\Delta w>0$, so the Hopf boundary lemma yields $w^{\prime }(r_{\ast })>0$, contradicting the first-contact requirement $w^{\prime }(r_{\ast })\leq 0$.
  • Figure 4: $u=y$ separates the regions $u>y$ and $u<y$. The trajectory starts in $u>y$ at $R_{0}$ and touches the diagonal for the first time at $r_{\ast }$. The maximum principle forbids crossing into $u<y$ here.

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4: Relation to Lair's result
  • Remark 2.5: Relation to Zhang et al.'s result
  • Remark 2.6
  • proof
  • proof
  • proof