Keller–Osserman Condition in Elliptic PDEs

• Keller–Osserman condition is a precise integral criterion that distinguishes between the existence of entire solutions and finite-domain blow-up in nonlinear elliptic equations.
• It provides critical growth thresholds for semilinear, quasilinear, and fully nonlinear operators, influencing both global behavior and boundary blow-up phenomena.
• Recent research extends the condition to elliptic systems, nonlocal contexts, and geometric settings, offering sharp quantitative estimates and asymptotic analyses.

The Keller–Osserman type condition is a fundamental criterion in the theory of nonlinear elliptic partial differential equations, characterizing the existence or nonexistence of entire or large solutions for a broad spectrum of semilinear, quasilinear, fully nonlinear, and system settings. Originating in seminal works of Keller and Osserman in the late 1950s, this condition formulates the precise growth thresholds required of the nonlinearity for global solvability versus finite-domain blow-up, and has been extended and adapted to a wide diversity of operator structures, geometric backgrounds, regularity regimes, and general sources.

1. Classical Keller–Osserman Integral and Semilinear Cases

The prototypical context is the semilinear elliptic equation

$\Delta u = f(u) \quad \text{in}~\mathbb{R}^{n}$

with $f \colon [0, \infty) \to [0, \infty)$ continuous and nondecreasing. The classical Keller–Osserman condition asserts that the existence of positive entire (global) solutions is governed by the integrability of

$\int_1^\infty \frac{ds}{\sqrt{F(s)}}$

where $F(s) = \int_0^s f(t)\;dt$ (Sirakov et al., 27 Oct 2025). If this integral diverges, entire solutions exist; convergence implies all solutions blow up at a finite radius. The same criterion describes boundary blow-up phenomena in bounded domains. The proof strategy utilizes the ODE reduction for radial solutions, barrier construction, and comparison principles.

The Keller–Osserman integral admits immediate extension to the nonhomogeneous Dirichlet boundary problem: $$- \operatorname{div}(A(x) \nabla u) + B(u(x)) = f(x), \quad u(x) \to +\infty \ \text{as}\ x \to \partial\Omega$$ where, in the homogeneous case $f(x) \equiv 0$, the condition reads

$$\int_{s_0}^\infty (2 G(s))^{-1/2} ds < \infty \quad \text{with} \quad G(s) = \int_0^s B(\tau) d\tau$$

(see (Diaz, 2022)). For power nonlinearities $B(u) = u^m$, the threshold is $m > 1$.

2. Systems and Generalized Nonlinearities: Keller–Osserman for Elliptic Systems

In semilinear and quasilinear elliptic systems—particularly competitive or cooperative systems of the form

$$\Delta u = p(r) f(u,v), \quad \Delta v = q(r) g(u,v)$$

with radial weights $p, q$, the criterion must account for the coupling structure. The modern approach introduces monotone envelope functions (“gauge” functions), such as $h(s+t)$, with $h$ continuous, nondecreasing, $h(0)=0$, and $f(s,t)+g(s,t) \geq h(s+t)$ for $s,t$ large (Covei, 15 Sep 2025, Covei, 4 Sep 2025).

The generalized Keller–Osserman integral condition becomes

$$\int_1^\infty \frac{ds}{\sqrt{\widetilde{H}(s)}} < \infty, \qquad \widetilde{H}(t) = \int_0^t h(z) dz$$

and, for systems, may involve coupled reciprocal integrals: $$\int_1^\infty \frac{dt}{g(f(t))} < \infty, \qquad \int_1^\infty \frac{dt}{f(g(t))} < \infty$$ as in systems with nonlinear right-hand sides depending on the other component (Covei, 4 Sep 2025). The threshold for largeness/boundedness may further depend on admissible sets of central values and auxiliary weight integrals. Precisely, these criteria delineate existence and classification of entire large, bounded, or semifinite solutions, as well as sharp quantitative growth estimates at infinity (Covei, 2015, Covei, 2016).

3. Quasilinear, Fully Nonlinear, and Higher-Order Extensions

For operators beyond the Laplacian, e.g., the $p$-Laplacian, Orlicz-Laplacian, or general degenerate/fully nonlinear operators, the Keller–Osserman condition retains its controlling role but must be adapted to the operator’s nonlinearity. In these settings, the divergence criterion typically becomes

$\int_1^{\infty} [F(s)]^{-1/p} ds = \infty \quad\text{(for %%%%12%%%%-Laplacian)}$

or, in an Orlicz–Sobolev context,

$\int_1^{\infty} [\Phi^{-1}(F(s))]^{-1} ds < \infty$

where $\Phi$ is the $N$-function for the Orlicz space (Santos et al., 2016, Magliaro et al., 2010, Morales, 2018). Formulations relying on sub- and supersolution construction, monotone iteration, and comparison principles are structurally robust.

For fully nonlinear second-order equations, including $k$-Hessian or Pucci-type operators, the generalized Keller–Osserman condition involves exponents dependent on the structural nonlinearity: $$\int^{\infty} \left( \int^t f(s)\, ds \right)^{-1/q} dt = \infty$$ where $q$ relates to the operator (e.g., $q=2$ for Laplacian, $q=k+1$ for $k$-Hessian) (Covei, 2015, Dolcetta et al., 2015, Li et al., 2022, Ji et al., 2022). For higher-order equations, explicit weight factors may appear (Kon'kov et al., 2018).

4. Keller–Osserman in Geometric, Nonlocal, and Variational Settings

Extensions to geometric or nonlocal contexts, such as equations on manifolds, in the Heisenberg group, or involving the fractional Laplacian, preserve the fundamental integrability motif, but add geometric or scaling corrections. On a Riemannian manifold, the condition becomes

$$\int^{\infty} K^{-1}(F(s)) ds < \infty, \qquad K(t) = \int_0^t \frac{s \varphi'(s)}{l(s)} ds$$

where $\varphi$ and $l$ encode the operator’s coercivity and diffusion structure (Bianchini et al., 2018, Magliaro et al., 2010). For fractional Laplacians, the scaling term changes; the “fractional Keller–Osserman” is

$$\int_1^\infty \left[\phi(t)\right]^{1/s} dt < \infty, \qquad \phi(t) = \int_t^\infty \frac{d\tau}{\sqrt{F(\tau)}}$$

with critical exponents reflecting the nonlocality (Abatangelo, 2014). In all cases, the existence of large solutions is tied to the divergence of the corresponding KO-type integral.

5. Applications, Proof Strategies, and Structural Insights

The Keller–Osserman framework enables:

• Characterization of existence/nonexistence (Liouville theorems, boundary blow-up profiles)
• Classification of solution behavior at infinity and near boundaries
• Quantitative asymptotics and sharp a priori bounds
• Delineation of critical and supercritical nonlinearity regimes (e.g., $f(u) \sim u^m$ with threshold $m > 1$ for Laplacian)
• Analysis in the presence of weights, lower-order terms, and unbounded coefficients, e.g., divergence-form operators with $L^q_{\text{uloc}}$ coefficients (Sirakov et al., 27 Oct 2025).

Proofs typically combine monotone iteration, barrier/subsolution construction via the integral transform (the “Keller–Osserman transform”), reduction to scalar or radial ODEs, and comparison principles. The essential observation is the equivalence between the divergence of an explicit integral and the existence of global solutions.

6. Generalized and Reciprocal Keller–Osserman Criteria for Systems

For semilinear systems with general nonlinearities and weights, the condition may involve envelopes such as

$$\int_1^{\infty} \frac{ds}{\sqrt{\int_0^s h(z) dz}} < \infty$$

for appropriate gauge $h$, or in coupled system settings,

$\int_1^{\infty} \frac{dt}{g(f(t))} < \infty$

together with matching conditions for each component, as prescribed by the system structure. For fully coupled systems, trichotomies of existence (both components large, one bounded/one large, both bounded) are classified by a combination of KO-type integrals and integrated weight constraints (Covei, 15 Sep 2025, Covei, 4 Sep 2025, Covei, 2015, Covei, 2016, Covei, 2011).

7. Criticality, Sharpness, and Open Directions

The Keller–Osserman condition is generally sharp: failure to satisfy the divergence criterion leads to nonexistence of entire solutions or boundedness. In each operator setting—the scalar, system, quasilinear, fully nonlinear, or geometric—the threshold captures the critical growth needed for largeness (blow-up) versus boundedness. Extensions to higher-order, weighted, nonlocal, and anisotropic operators remain active areas, with current research refining the interplay between scaling, geometry, and nonlinearity (Diaz, 2022, Diaz, 2022, Li et al., 2022).

Summary Table: Prototypical Keller–Osserman Conditions

Setting	KO-type Integral	Existence ⇔ (divergence of KO)
$\Delta u = f(u)$	$\int_1^\infty [F(s)]^{-1/2} ds$	Yes
$-\operatorname{div}(\phi(|\nabla u|)\nabla u)$	$\int_1^{\infty} [\Phi^{-1}(F(s))]^{-1} ds$	Yes
$p$-Laplacian, $\Delta_p u = f(u)$	$\int_1^\infty [F(s)]^{-1/p} ds$	Yes
Systems (two components)	Coupled integrals (e.g. $\int_1^\infty ds/\sqrt{H(s)}$ or reciprocal forms)	Yes (conditional on both components)
$k$-Hessian/f. nonlinear	$\int^\infty \left(\int^t f^k \right)^{-1/(k+1)} dt$	Yes
Fractional Laplacian $(-\Delta)^s$	$\int_1^\infty [\phi(t)]^{1/s} dt$, $\phi(t)=\int_t^\infty d\tau/\sqrt{F(\tau)}$	Yes

The Keller–Osserman condition thus provides a unifying and robust criterion for the qualitative analysis of nonlinear elliptic boundary value problems, fully determining the existence and large solution regimes across a wide range of analytic frameworks and mathematical models. For recent generalizations and optimal extensions to unbounded coefficients and irregular settings, see (Sirakov et al., 27 Oct 2025, Covei, 15 Sep 2025, Covei, 4 Sep 2025), and (Diaz, 2022).