Critical Quasilinear Equations

• Quasilinear equations with critical nonlinear terms are PDEs characterized by degenerate or anisotropic operators and nonlinearities that meet sharp embedding thresholds.
• They pose challenges in regularity, existence, and multiplicity analyses, often necessitating advanced variational methods such as mountain pass and concentration–compactness principles.
• Practical applications span nonlinear optics, heterogeneous diffusion, and pattern formation, linking theoretical insights with real-world phenomena.

Quasilinear equations with critical nonlinear terms are partial differential equations (PDEs) in which the principal part is quasilinear—typically of degenerate or anisotropic type—and the nonlinearity exhibits “critical” growth as specified by the invariance of associated Sobolev or Orlicz–Sobolev embeddings, homogeneous scaling, Trudinger–Moser type exponential rates, or other natural thresholds arising from the geometry or function space context. These problems are characterized by an intricate interplay between the structure of the differential operator and the sharpness or borderline nature of the nonlinear terms, resulting in delicate regularity, existence, compactness, and multiplicity phenomena.

1. Critical Nonlinearities in Quasilinear Frameworks

Critical nonlinearities are those whose growth matches, in the appropriate sense, the invariance or optimality threshold of a relevant embedding. For the $p$-Laplacian, this occurs at the Sobolev critical exponent $p^* = np/(n-p)$, while for operators on Carnot groups, Grushin-type operators, or in variable exponent settings, the exponent is computed in terms of the homogeneous or effective dimension. The archetypal critical quasilinear elliptic equation in divergence form is

$$-\Delta_p u = |u|^{p^*-2}u \quad \text{in } \Omega \subset \mathbb{R}^n, \quad u=0 \text{ on } \partial\Omega,$$

while more general settings include

$$-\operatorname{div}(\phi(|\nabla u|)\nabla u) = \lambda \phi^*(|u|)u + f(x,u),$$

with the nonlinearity $f$ attaining critical Orlicz or Trudinger–Moser growth, and equations with lower order terms exhibiting natural or superlinear growth in $u$ or $|\nabla u|$ (Jaye et al., 2010, Santos, 2015, Ochoa et al., 2020, Biswas et al., 2021, Biswas et al., 2022).

Criticality may also manifest in variable exponent spaces, e.g., for the $p(x)$-Laplacian, or through nonlocal and convolution-type terms, as in Choquard-type or Hénon-type equations featuring critical exponential rates (Pomponio et al., 2020, Biswas et al., 2021).

2. Examples: Classical and Generalized Critical Models

Research has established a wealth of models illustrating the subtlety of critical nonlinearities in quasilinear settings:

• p-Laplace equations with critical Sobolev exponent:

$$-\Delta_p u = u^{p^*-1} \quad \text{in } \mathbb{R}^n,$$

whose solutions include Aubin–Talenti bubbles exhibiting precise asymptotics and loss of compactness due to dilation invariance (Sun et al., 12 Feb 2025, Clapp et al., 25 Feb 2025, Gandal et al., 7 Sep 2025).

• Quasilinear operators with critical lower-order terms:

$$-\operatorname{div}(|\nabla u|^{p-2}\nabla u) = \sigma(x)u^{p-1} + \delta_{x_0},$$

where the potential $\sigma$ is at the threshold for capacity smallness, engendering bilateral exponential global bounds for the fundamental solution (Jaye et al., 2010).

• Degenerate and subelliptic models (Grushin operators):

$$-\Delta_{\gamma,p} u = \lambda |u|^{q-2}u + |u|^{p_\gamma^*-2}u,$$

with the Grushin gradient and associated homogeneous dimension $N_\gamma$ governing the critical exponent $p_\gamma^*$ (Gandal et al., 7 Sep 2025).

• Variable exponent and anisotropic frameworks:

Incorporation of $p(x)$-growth, or operators weighted by anisotropic matrices, introduces new critical phenomena tied to variable geometry or media (Ochoa et al., 2020, Gluck, 2023).

• Critical exponential or Trudinger–Moser-type nonlinearities:

Quasilinear problems in low dimension (e.g., $N=2$) frequently display critical growth of exponential type,

$f(x,s) \sim \exp(|s|^\alpha),$

for $\alpha$ at the Moser threshold, as in modified quasilinear Schrödinger and strongly singular equations (Biswas et al., 2022, Biswas et al., 2021).

3. Variational Structure, Compactness, and Threshold Phenomena

The critical growth regime is marked by the breakdown of compactness for embedding theorems, directly impacting the calculus of variations approach:

• Sharp Sobolev and Orlicz–Sobolev inequalities: Existence of extremal functions (minimizers for the associated inequalities) becomes a crucial step in constructing solutions on unbounded or degenerate domains (Gandal et al., 7 Sep 2025, Santos, 2015).
• Mountain Pass and genus theory: Critical nonlinearity often induces a noncompact Palais–Smale sequence, so tools like the mountain pass theorem, concentration–compactness principle, and genus-theoretic minimax schemes are adapted for quasilinear settings (Santos, 2015, Clapp et al., 25 Feb 2025).
• Existence versus nonexistence and multiplicity: The presence of a critical term often induces strict thresholds depending on geometric, spectral, or topological characteristics of the domain or coefficients. For example, in Brezis–Nirenberg-type results, the existence of solutions for the critical $p$-Laplacian is linked to the relative size of domain, spectral parameters (first eigenvalue), and analytic invariants (Gandal et al., 7 Sep 2025, Clapp et al., 26 Feb 2025).
• Symmetry-breaking and nodal solutions: Criticality, combined with symmetries of the domain or operator, can lead to the existence of entire, nodal, or "pinwheel" solutions, and to multiplicity phenomena absent from the subcritical regime (Clapp et al., 25 Feb 2025, Gluck, 2023).

4. Regularity Theory and Sharp Estimates

Quasilinear equations with critical nonlinearities pose severe analytic challenges near degeneracies or singularities. Recent advances have yielded:

• Interior and fine regularity: Under suitable smallness or continuity assumptions on lower-order data, one obtains $C^{1,\alpha}$ interior regularity for viscosity solutions even in nondivergence form, with estimates sharp at critical points where the operator degenerates or the gradient vanishes (Bessa et al., 14 Apr 2025).
• Quantitative behavior near critical points: Precise growth and separation results at extrema (non-degeneracy) are established, ruling out excessive vanishing and giving control over interfaces or free boundaries.
• Strong maximum principle and Hopf lemma: These classical tools are extended to degenerate, quasilinear, and nondivergence structures, with the maximum principle applying despite strong nonuniformity at critical points (Bessa et al., 14 Apr 2025).
• Gradient and Kato-type inequalities on manifolds: For geometric settings or anisotropic media, sharp nonlinear Kato inequalities and gradient Cheng–Yau estimates underpin rigidity and classification results for critical equations, connecting regularity to curvature and ambient geometry (Sun et al., 12 Feb 2025).

5. Nonlinear Potential Theory and Capacitary Methods

The equilibrium between quasilinear operators and critical (natural growth) nonlinearities frequently requires tools from nonlinear potential theory:

• Wolff potentials and nonlinear integral operators: These capture the precise manner in which singular measures or small-scale features of the lower-order term modulate fundamental solutions, as in the two-sided global estimates for equations perturbed by critical measures (Jaye et al., 2010).
• Weighted norm inequalities and capacities: For Riccati type quasilinear equations, existence and a priori estimates are secured via nonlinear Muckenhoupt–Wheeden bounds and capacitary smallness conditions, generalizing classical weighted analysis to the quasilinear and critical context (Phuc, 2013).
• Blow-up and Liouville-type classification: Iterative, rescaling, and blow-up arguments, combined with Liouville theorems, preclude nontrivial entire solutions in subcritical regimes or characterize solution profiles at criticality (Chang et al., 2020, Sun et al., 12 Feb 2025).

6. Applications, Extensions, and Directions

Quasilinear equations with critical nonlinear terms continually arise in physical and geometric contexts:

• Nonlinear optics and field theory: Critical quasilinear scalar field or Schrödinger equations naturally appear in models of self-trapped beams, high-power light propagation, and nonlinear fluids, especially when critical exponents control the qualitative behavior of the response (Pomponio et al., 2020, Liu et al., 2022, Deng et al., 2 Mar 2024).
• Diffusion in heterogeneous and degenerate media: Grushin-type and degenerate operators describe phenomena where ellipticity fails along lower-dimensional sets, demanding refined functional analytic and variational approaches (Gandal et al., 7 Sep 2025).
• Competitive systems and pattern formation: Purely critical systems for the $p$-Laplacian display intricate solution structures, including entire, sign-changing, and symmetry-constrained states (Clapp et al., 25 Feb 2025).
• Nonlocal, singular, or multiplicative nonlinearities: Problems with critical nonlocal, Choquard, or sign-changing (sharp interface) nonlinearities exhibit threshold and multiplicity phenomena strongly influenced by domain topology and the interaction between singular terms and criticality (Biswas et al., 2021, Biswas et al., 2022, Clapp et al., 26 Feb 2025).

Quasilinear problems with critical nonlinearities thus form a nexus between nonlinear analysis, geometry, and applied PDEs. Their paper continues to motivate methodological innovations in regularity theory, nonlinear potential analysis, functional inequalities, and variational techniques, as well as to uncover new phenomena in mathematical physics, geometry, and materials science.