Keller–Osserman Condition in Nonlinear PDEs

• Keller–Osserman condition is a fundamental integral criterion that defines when nonlinear PDEs admit large solutions with boundary or infinite blow-up.
• It generalizes across semilinear, quasilinear, Orlicz, Hessian, and fractional operators by adjusting integrability conditions for various nonlinearities.
• Analytic techniques such as radial ODE reduction, barrier construction, and energy methods solidify its sharp threshold and broad applicability.

The Keller–Osserman condition is a fundamental integral criterion that controls the existence and qualitative behavior of "large solutions"—solutions exhibiting blow-up at the boundary or at infinity—for a wide class of nonlinear partial differential equations (PDEs) and systems, including semilinear, quasilinear, fully nonlinear, and higher-order models. The core principle is that the growth of the source term or nonlinearity must be sufficiently strong to permit solutions that escape to infinity, and the precise analytic threshold is encoded in the integrability properties of an associated functional. Originally formulated for scalar semilinear elliptic equations, the Keller–Osserman condition has since been generalized across an extensive landscape of PDEs, including systems, degenerate and fully nonlinear operators, Hessian equations, equations with weights and gradient dependence, on manifolds, and for nonlocal (fractional Laplacian) operators.

1. Classical Statement and Variants

The classical Keller–Osserman condition, established for the autonomous semilinear elliptic equation

$$\Delta u = f(u) \quad \text{in} \quad \Omega \subseteq \mathbb{R}^N, \quad u(x) \to +\infty \quad \text{as} \quad x \to \partial\Omega$$

(where $f:[0,\infty)\to[0,\infty)$ is continuous and nondecreasing), asserts that a necessary and sufficient condition for the existence of at least one positive solution with boundary blow-up is

$$\int_{s_0}^{+\infty} \frac{dt}{\sqrt{F(t)}} < +\infty, \qquad F(t) := \int_0^t f(s)\,ds.$$

This criterion also appears, with variations in exponents and inner integrands, in more general cases:

• For the $p$-Laplacian, it reads $\int^{\infty} dt / [F(t)]^{1/p} < \infty$.
• For fully nonlinear degenerate elliptic equations, the same principal form applies as a threshold for entire viscosity subsolutions (Dolcetta et al., 2015).
• For boundary blow-up in bounded domains with nonuniform ellipticity, the criterion remains unchanged provided the operator is uniformly elliptic in the normal direction near ∂Ω (Diaz, 2022).
• For higher-order equations, the threshold takes the form $\int_1^\infty g(s)^{-1/m} s^{1/m - 1} ds < \infty$, where $g$ is the nonlinearity and $m$ is the operator order (Kon'kov et al., 2018).

2. Generalizations to Quasilinear, Orlicz, and Fully Nonlinear Operators

The condition is robust under significant elaboration:

• Quasilinear and Orlicz Settings: For operators in divergence form $\mathrm{div}(\phi(|\nabla u|)\nabla u)$, the generalized Keller–Osserman condition becomes

$\int^\infty \frac{dt}{\Phi^{-1}(F(t))} < \infty,$

where $f:[0,\infty)\to[0,\infty)$0 and $f:[0,\infty)\to[0,\infty)$1 is the generalized inverse (Santos et al., 2016).

• Hessian and k-Hessian Equations: In equations involving the k-Hessian operator $f:[0,\infty)\to[0,\infty)$2, the natural generalization is

$f:[0,\infty)\to[0,\infty)$3

with the geometric requirement that the domain be strictly (k–1)-convex (Covei, 2015, Li et al., 2022).

• Operators with Gradient Terms: When gradient dependence is present, e.g., in $f:[0,\infty)\to[0,\infty)$4, the condition adapts via a change of gauge determined by the interplay of the function $f:[0,\infty)\to[0,\infty)$5 and the weight $f:[0,\infty)\to[0,\infty)$6, resulting in integrals of the form $f:[0,\infty)\to[0,\infty)$7, where $f:[0,\infty)\to[0,\infty)$8 encodes the operator structure (Magliaro et al., 2010, Bianchini et al., 2018).

3. Keller–Osserman Conditions for Systems

A substantial extension of the theory addresses elliptic systems, particularly of semilinear and quasilinear type:

• Diagonal Systems and Systems with Weights: For two-equation systems such as

$f:[0,\infty)\to[0,\infty)$9

existence and (large/bounded) qualitative behavior depend on componentwise Diagonal KO Integrals:

$$\int_{s_0}^{+\infty} \frac{dt}{\sqrt{F(t)}} < +\infty, \qquad F(t) := \int_0^t f(s)\,ds.$$0

and on the finiteness or divergence of associated weight integrals $$\int_{s_0}^{+\infty} \frac{dt}{\sqrt{F(t)}} < +\infty, \qquad F(t) := \int_0^t f(s)\,ds.$$1 (Covei, 2016, Covei, 2015).

• General Nonlinearities via Envelope Functions: For non-power-type or competitive systems, a univariate envelope function $$\int_{s_0}^{+\infty} \frac{dt}{\sqrt{F(t)}} < +\infty, \qquad F(t) := \int_0^t f(s)\,ds.$$2 is constructed such that the system’s effective nonlinearity is bounded below by $$\int_{s_0}^{+\infty} \frac{dt}{\sqrt{F(t)}} < +\infty, \qquad F(t) := \int_0^t f(s)\,ds.$$3. The existence of large solutions is then governed by

$$\int_{s_0}^{+\infty} \frac{dt}{\sqrt{F(t)}} < +\infty, \qquad F(t) := \int_0^t f(s)\,ds.$$4

with further conditions specifying whether both or only one component blows up (Covei, 15 Sep 2025).

• Reciprocal (Composite) Integral Conditions: For coupled systems of the type

$$\int_{s_0}^{+\infty} \frac{dt}{\sqrt{F(t)}} < +\infty, \qquad F(t) := \int_0^t f(s)\,ds.$$5

the decisive integral conditions become

$$\int_{s_0}^{+\infty} \frac{dt}{\sqrt{F(t)}} < +\infty, \qquad F(t) := \int_0^t f(s)\,ds.$$6

capturing the mutual amplification effects of the system (Covei, 4 Sep 2025).

Equation/System Type	Associated KO-type Integral(s)	Parameter/Structural Dependence
Scalar semilinear	$$\int_{s_0}^{+\infty} \frac{dt}{\sqrt{F(t)}} < +\infty, \qquad F(t) := \int_0^t f(s)\,ds.$$7	$$\int_{s_0}^{+\infty} \frac{dt}{\sqrt{F(t)}} < +\infty, \qquad F(t) := \int_0^t f(s)\,ds.$$8
$$\int_{s_0}^{+\infty} \frac{dt}{\sqrt{F(t)}} < +\infty, \qquad F(t) := \int_0^t f(s)\,ds.$$9-Laplace/Orlicz	$p$0	$p$1
Hessian ($p$2-Hessian)	$p$3	Strict (k–1)-convex domain
Systems with diagonals	$p$4	Each component analyzed separately
Coupled/reciprocal systems	$p$5, $p$6	Mutual compositions
General nonlinearities	$p$7	Envelope function $p$8

4. Proof Strategies and Analytic Mechanisms

Across all settings, the establishment of the Keller–Osserman phenomenon relies on a unifying set of analytic tools:

• Reduction to Radial ODE and Construction of Barriers: The mechanics usually begin with reduction to a radial ODE governing the blow-up. Integrability conditions determine whether this ODE admits entire or boundary-blow-up solutions (e.g., via double integration for the k-Hessian operator or following similar schemes for systems) (Covei, 2015).
• Monotone Iteration and Comparison Principle: The method of sub and supersolutions, coupled with monotone iteration (possibly exploiting convexity or monotonicity of the nonlinearity) is used to construct global solutions and establish minimality/maximality or uniqueness when available (Covei, 2016, Covei, 15 Sep 2025).
• Integral Transforms and ODE Energy Techniques: Keller–Osserman–type transforms (e.g., $p$9 for systems, or envelope-based transforms) are employed to control growth and obtain a priori bounds or sharp asymptotics (Covei, 15 Sep 2025, Covei, 4 Sep 2025).
• Handling Degenerate Operators/Nonlocality: For degenerate or nonlocal operators, such as the fractional Laplacian, KO conditions are strengthened and may involve more stringent (often doubled) integrability against the nonlinear primitive or its inverse (Abatangelo, 2014).

5. Extensions: Geometry, Operators with Weights, and Manifolds

• Weighted and Anisotropic Operators: The presence of nonconstant weights in divergence-form operators (e.g., $\int^{\infty} dt / [F(t)]^{1/p} < \infty$0) does not alter the fundamental integral criterion, provided the lower order coefficients have sufficient local integrability (e.g., in $\int^{\infty} dt / [F(t)]^{1/p} < \infty$1) (Sirakov et al., 27 Oct 2025).
• Manifold Settings and Geometric Thresholds: On noncompact Riemannian manifolds, the Keller–Osserman condition characterizes validity of the strong maximum principle, Liouville property, and compact support principle, interacting subtly with geometric features such as Ricci curvature bounds and volume growth rates (Bianchini et al., 2018).
• Nonlocal and Fractional Operators: For equations driven by the fractional Laplacian, the classical Keller–Osserman condition is reinforced by a higher integrability exponent and, in some cases, an additional moment condition (e.g., $\int^{\infty} dt / [F(t)]^{1/p} < \infty$2) (Abatangelo, 2014).

6. Optimality, Sharpness, and Constructive Counterexamples

The Keller–Osserman threshold is universally sharp. Whenever the associated integral diverges, the existence of entire or blow-up solutions can be excluded, while convergence guarantees existence, often with explicit or barrier-construction exhibiting the minimal blow-up profile. This principle holds uniformly in all generalizations, including:

• Degenerate/fully nonlinear operators (Dolcetta et al., 2015),
• Higher-order inequalities (Kon'kov et al., 2018),
• Quasilinear systems with absorption/mixed terms (with explicit Harnack behaviour in some components) (Bidaut-Véron et al., 2011).

Sharpness is further illuminated through explicit examples: for power-law nonlinearities, the critical exponent aligns precisely with the divergence of the Keller–Osserman integrals; for logarithmic nonlinearities at threshold, only the borderline cases may admit nontrivial global solutions.

7. Impact and Unifying Role in Nonlinear PDE Theory

The Keller–Osserman condition serves as a universal threshold in nonlinear potential theory, controlling the dichotomy between the existence and nonexistence of boundary blow-up and large solutions for nonlinear elliptic (and sometimes parabolic) problems, both scalar and vectorial. There are no known alternative analytic criteria of comparable generality for these phenomena. The methodology has been substantially extended via monotone envelope transforms and reciprocal integral conditions, providing a systematic device for verifying large solution behaviour in increasingly sophisticated operator, geometry, and system settings (Covei, 15 Sep 2025, Covei, 2015, Bianchini et al., 2018).