Boundary Blow-Up Problem Overview

• Boundary blow-up problems are nonlinear PDEs where solutions become unbounded at the domain's boundary while remaining finite in the interior.
• They are analyzed using methods such as sub- and super-solution techniques, radial ODE reduction, and variational iterations to establish existence and multiplicity.
• Precise asymptotic analysis, notably via Keller–Osserman conditions, is key to understanding the unique blow-up profiles and their sensitivity to operator types and boundary data.

A boundary blow-up problem refers to a class of nonlinear partial differential equations (PDEs) or systems in which solutions become unbounded ("blow up") as one approaches the boundary of the domain, while remaining finite in the interior. This phenomenon arises in semilinear, quasilinear, fully nonlinear, nonlocal, and geometric settings, and often requires refined methods to analyze existence, uniqueness, multiplicity, and precise asymptotics near the boundary.

1. Core Formulations and Prototypical Equations

The prototype for a boundary blow-up problem is the elliptic equation or system, posed in a smooth bounded domain $\Omega\subset\mathbb{R}^N$, with the "large solution" boundary condition. A representative example is

$$\begin{cases} -\Delta u = f(u) & \text{in } \Omega \ u(x) \to +\infty & \text{as } \mathrm{dist}(x,\partial\Omega)\to0 \end{cases}$$

where $f$ is a nondecreasing nonlinearity. The requirement $u\to+\infty$ near $\partial\Omega$ is the hallmark of the blow-up regime. The p-Laplacian, Monge-Ampère, and fractional Laplacian settings generalize this by introducing non-standard diffusion or nonlocality:

• $-\Delta_p u = f(u)$ with $\Delta_p u = \nabla\cdot(|\nabla u|^{p-2}\nabla u)$;
• Nonlocal: $(-\Delta)^{\frac{\alpha}{2}} u = f(u)$ for $0<\alpha<2$;
• Fully nonlinear real Monge-Ampère operators such as $\det^{1/(N-1)}(\Delta z\,I-D^2 z)=K(|x|)f(z)$ (Saha et al., 15 Nov 2025);
• Quasilinear problems with logistic or singular nonlinearity (Alves et al., 2015).

Boundary blow-up phenomena also appear in geometric PDEs involving conformal deformations of scalar and mean curvatures, e.g., the boundary Yamabe problem (Cruz-Blázquez et al., 2023, Cruz-Blázquez, 2023, Ghimenti et al., 2018), and in parabolic or free boundary problems (Gladkov, 2016, Badger et al., 2022).

2. Analytical Techniques and Existence Theory

The existence and multiplicity of boundary blow-up solutions are typically established using a combination of:

• Sub- and super-solution methods: Construction of explicit barriers or comparison profiles that are ordered and which enable application of comparison principles (Saha et al., 15 Nov 2025, Alves et al., 2015, Chrouda et al., 2015, Gladkov, 2016).
• Reduction to radial ODEs: In symmetric domains (e.g., balls), assuming radial symmetry reduces high-dimensional PDEs to singular ODEs with blow-up at the endpoint (Saha et al., 15 Nov 2025).
• Monotone/variational iterations: Variational minimization between barriers or monotone sequences, often passing through truncated problems with large Dirichlet boundary data before taking limits (Alves et al., 2015, Chrouda et al., 2015).
• Shooting method/parameter continuation: Continuation in initial values of ODE reductions yields infinite families of blow-up solutions under appropriate monotonicity and growth conditions (Saha et al., 15 Nov 2025, Inaba et al., 29 May 2024).
• Lyapunov-Schmidt reduction: In geometric and critical problems, solutions are constructed as perturbations of model bubbles, reducing to finite-dimensional variational problems governing concentration points or blow-up rates (Ghimenti et al., 2018, Cruz-Blázquez et al., 2023, Cruz-Blázquez, 2023).

A central analytical motif is the adaptation of the Keller–Osserman condition: $\int^{\infty} [F(s)]^{-\beta}ds = \infty$ for suitable $\beta>0$ and $F(s)=\int_0^s f(t)dt$. This condition determines the threshold for the existence of boundary blow-up solutions in both classical and fully nonlinear equations; the exponent $\beta$ depends on the operator structure (Saha et al., 15 Nov 2025, Alves et al., 2015, Chrouda et al., 2015).

3. Asymptotics and Uniqueness of the Blow-Up Profile

The precise rate at which solutions diverge near the boundary is a central issue. For semilinear and quasilinear elliptic equations, the leading order blow-up profile is typically of the form

$$u(x) \sim A(x_0)d(x)^{-\alpha(x_0)}, \quad \text{as } x\to x_0\in\partial\Omega$$

with $d(x)=\mathrm{dist}(x,\partial\Omega)$. Here, $\alpha(x_0)$ and $A(x_0)$ are determined by a balance between the leading order terms in the equation and the local geometry or weight functions at the boundary, often via reduction to boundary-normal or Fermi coordinates and scalings (Alves et al., 2015, Cruz-Blázquez et al., 2023, Cruz-Blázquez, 2023, Ghimenti et al., 2018). In nonlocal fractional Laplacians, the exponent is shifted relative to the local case and the classical Keller–Osserman threshold fails (Chrouda et al., 2015).

Uniqueness is subtle and depends on the regime and operator type. While uniqueness can hold when comparison principles are strong and the boundary singularity is isolated and monotonicity conditions are met (Alves et al., 2015), multiplicity may arise:

• In the (N-1)-Monge-Ampère problem with singular weights and Keller–Osserman growth, infinitely many radial solutions are constructed via varying initial conditions in the shooting method (Saha et al., 15 Nov 2025).
• In nonlocal or quasilinear ODEs with nonlocal coupling, bifurcation diagrams with multiple blow-up branches can occur (Inaba et al., 29 May 2024).

In geometric problems (e.g., Yamabe), the localization and scaling of the "bubble" is determined by maximization of reduced energies, with nondegenerate critical points of curvature functions determining the blow-up location (Cruz-Blázquez et al., 2023, Cruz-Blázquez, 2023, Ghimenti et al., 2018).

4. Extensions: Nonlocal, Parabolic, and Free Boundary Regimes

Boundary blow-up is broadly encountered beyond elliptic settings:

• Parabolic equations with nonlinear nonlocal boundary coupling: Blow-up may be driven either by bulk reaction terms or by supercritical boundary fluxes, with explicit criteria delineating global existence and finite-time blow-up regimes (Gladkov, 2022, Gladkov, 2016). In these settings, blow-up may be localized solely on the boundary, as shown via representation formulae and interior kernel bounds (Gladkov, 2016).
• Fractional order equations: The threshold for large solution existence shifts upward in the nonlocal regime, invalidating the direct transfer of local theory and requiring fractional potential analysis and Martin kernel expansions (Chrouda et al., 2015).
• Nonlocal boundary blow-up ODEs: The addition of nonlocal coefficients coupling to the global norm of the solution introduces novel bifurcation phenomena including fold bifurcations, with sharp asymptotic expansions for distinct solution branches as parameters vary (Inaba et al., 29 May 2024).

In free boundary problems, blow-up analysis classifies possible tangent cones and singular sets. For regular data, the boundary blow-up limit is unique and corresponds to a canonical harmonic polynomial, while for merely continuous coefficients, non-uniqueness and a rich "moduli" of tangent cones arise (Badger et al., 2022, Minne, 2015).

5. Geometric and Variational Contexts

Blow-up behavior at the boundary is fundamental in geometrical problems involving conformal geometry, prescribed scalar and boundary mean curvature, and critical nonlinearities. In the boundary Yamabe problem and its variants (Ghimenti et al., 2018, Cruz-Blázquez et al., 2023, Cruz-Blázquez, 2023), blow-up solutions are constructed at elliptic boundary points, governed by curvature data and critical points of the mean curvature. Singular perturbation and Lyapunov–Schmidt reduction yield sharp asymptotic expansions for solution concentration, with distinguished geometric invariants (e.g., Weyl tensor, second fundamental form) dictating the possibility and location of blow-up. The interplay of interior and boundary critical nonlinearities requires careful analysis of the reduced energy landscape and its critical points, ensuring the isolation and stability of blow-up under parameter limits.

Boundary blow-up also plays a central role in sharp inequalities (e.g., the Trudinger–Moser inequality for surfaces with boundary), where blow-up analysis clarifies the attainment of extremals and the obstruction to compactness (Yang et al., 2020).

6. Summary Table: Operator Types and Blow-Up Features

Operator/Class	Existence Criteria	Asymptotics (Generic)	Uniqueness/Multiplicity
Semilinear, $\Delta u=f(u)$	Keller–Osserman condition	$u\sim C d(x)^{-\alpha}$	Often unique
p-Laplacian, $-\Delta_p u$	Modified Keller–Osserman	$u\sim A(x_0) d(x)^{-\alpha}$	Unique under monotonicity
(N-1)-Monge-Ampère	Integral growth, singular $K$	Varies, shooting construction	Infinitely many radial solutions
Nonlocal/fractional	Shifted critical exponent	$u\sim C d(x)^{-\beta}$	As above, non-uniqueness possible
Yamabe-type geometric	Energy max at critical point	Bubble: $C\varepsilon^{-(n-2)/2}$	Unique or isolated
Parabolic	Exponent balance, kernel cond.	Local/integral ODE rates	Non-uniqueness, localization

The precise nature of boundary blow-up is thus dictated by both analytic properties of the operator and the fine structure of the boundary data or weights. Across semilinear, quasilinear, nonlocal, and geometric PDEs, boundary blow-up solutions exemplify the interplay of singular analysis, comparison techniques, variational principles, and geometric invariants.

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References

• Real $(N-1)$-Monge-Ampère equations: existence of infinitely many radial blow-up solutions with singular weights and Keller–Osserman condition (Saha et al., 15 Nov 2025)
• Slowly vanishing mean oscillations and non-uniqueness of blow-up limits in free boundary problems (Badger et al., 2022)
• Quasilinear $p$-Laplacian blow-up: existence, uniqueness, and sharp boundary rates (Alves et al., 2015)
• Nonlocal and parabolic blow-up with nonlocal boundary conditions: comparison, localization, and bifurcation (Gladkov, 2016, Gladkov, 2022, Inaba et al., 29 May 2024)
• Semilinear fractional equations: threshold for large solutions and failure of classical criterion (Chrouda et al., 2015)
• Blow-up in geometric boundary value problems: Yamabe, critical exponents, curvature concentration (Ghimenti et al., 2018, Cruz-Blázquez et al., 2023, Cruz-Blázquez, 2023)
• Classification and uniqueness of blow-up polynomials at singular points for free boundary problems (Minne, 2015)
• Sharp Trudinger-Moser inequalities and boundary blow-up in variational problems on Riemann surfaces (Yang et al., 2020)