Nonlocal Critical-Power Nonlinearity

• Nonlocal critical-power nonlinearity is characterized by equations featuring critical exponents and power-dependent bifurcations that trigger sharp threshold transitions in solution behavior.
• Analytical tools such as Nehari manifold methods, fiber map analysis, and concentration-compactness are employed to manage loss of compactness in these nonlocal and critical settings.
• Applications span nonlinear optics and anomalous diffusion, where power thresholds govern phenomena like soliton ejection, Anderson localization, and ground state transitions.

Nonlocal critical-power nonlinearity refers to phenomena and mathematical structures where the nonlinearity present in an equation either attains the “critical” exponent for the associated (possibly nonlocal) Sobolev embedding, or where the system exhibits power-dependent bifurcations in the presence of nonlocal operators or integral coupling. This concept is central to the paper of nonlinear PDEs with fractional or nonlocal operators, nonlinear wave or optics equations with nonlocal responses, and systems where solution behavior undergoes threshold transitions as the power or amplitude crosses a critical value.

1. Mathematical Models Incorporating Nonlocal Critical-Power Nonlinearity

Nonlocal critical-power nonlinearities arise in equations where the nonlinear, often power-law, term is coupled to integral or fractional-differential operators. Representative models include:

• Nonlinear Schrödinger equations with nonlocal response, such as:

$$i\frac{\partial q}{\partial \xi} = -\frac{1}{2} \frac{\partial^2 q}{\partial \eta^2} - qn - pV(\eta - d)q \qquad \frac{\partial^2 n}{\partial \eta^2} = -|q|^2$$

The refractive index change $n$ depends nonlocally on the field $q$ (Ye et al., 2010).

• Nonlocal nonlinear Schrödinger equations with convolutional nonlinearity:

$$i \psi_t + \psi_{xx} = V(x)\psi - s\psi \int_{\mathbb{R}} \chi(x'-x) |\psi(x')|^2 dx'$$

Here, the response function $\chi$ defines the nonlocal response range (Folli et al., 2012).

• Nonlocal/fractional Laplacian equations with critical nonlinearity:

$$(-\Delta)^s u = u^{2^*_s-1}\quad \text{in}\;\Omega, \quad u = 0 \;\; \text{in} \; \mathbb{R}^N \setminus \Omega$$

where $2^*_s = 2N/(N-2s)$ is the fractional critical exponent (Mosconi et al., 2015).

• Nonlocal Kirchhoff-type or mixed local-nonlocal operators, as in:

$$([u]_{s,p}^p)^{\sigma-1}(-\Delta)^s_p u = \lambda u^{-\gamma} + u^{p_s^*-1}$$

(Ghanmi et al., 2022), and

$$(-\Delta_p) u + (-\Delta_q)^s u = \lambda a(x)|u|^{\delta-2}u + b(x)|u|^{r-2}u$$

(Dhanya et al., 1 Mar 2025).

These models typically exhibit the interplay between critical exponents (governing the threshold for compactness), nonlocal spatial interactions, and bifurcation phenomena in solution sets as the power crosses a critical threshold.

2. Critical Exponents and Their Role in Nonlocal Problems

In the context of nonlocal equations, a nonlinearity is called “critical” when its power matches the borderline of the Sobolev (or related) embedding:

• For the fractional Laplacian $(-\Delta)^s$ in $\mathbb{R}^N$, the critical exponent is

$2^*_s = \frac{2N}{N-2s}$

so the nonlinearity $u^{2^*_s-1}$ is critical for the embedding $$H^s(\mathbb{R}^N) \hookrightarrow L^{2^*_s}(\mathbb{R}^N)$$ (Mosconi et al., 2015).

• In convolution-type (Choquard or Hartree-type) equations, the critical exponent is determined via the Hardy–Littlewood–Sobolev (HLS) inequality:

$2^*_\mu = \frac{2n - \mu}{n - 2s}$

Here, nonlinearities of the type:

$$\left(\int_\Omega \frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu}dy\right)|u(x)|^{2^*_\mu - 2}u(x)$$

are called critical (Giacomoni et al., 2017, Anthal et al., 2023).

• For mixed (local/nonlocal) problems, the critical powers are given by

$$p^* = \frac{Np}{N-p}, \qquad p^*_s = \frac{Np}{N-ps}$$

with the nonlocal/fractional exponents controlling the compactness threshold (Ghanmi et al., 2022, Dhanya et al., 1 Mar 2025).

Critical growth causes a loss of compactness in variational settings, leading to phenomena such as energy concentration (“bubbling”), resonance behavior, and, in optics or nonlinear wave propagation, sharp thresholds for the existence or loss of soliton (self-trapped) states.

3. Power-Dependent Bifurcations and Threshold Effects

A defining feature of nonlocal critical-power nonlinearity is the presence of strict power thresholds that demarcate qualitative changes in solutions:

• Soliton ejection in nonlocal media: In nonlocal thermal media with a trapping defect, there exists a critical energy flow $U_{\max}$: for $U < U_{\max}$, the soliton remains trapped; for $U > U_{\max}$, nonlocal (power-dependent) deflection forces dominate, leading to ejection from the trap. $U_{\max}$ itself depends on defect strength and geometry (Ye et al., 2010).
• Anderson localization and delocalization: In highly nonlocal media, Anderson localized states become robust with respect to nonlinearity, but a critical power $P_c$ is necessary to destabilize or modify the localization. $P_c$ increases with the degree of nonlocality, scaling inversely with the overlap between the response function and the localized state (Folli et al., 2012).
• Ground state existence in mass-critical NLS: On certain nonlocal structures (e.g., doubly periodic metric graphs), the existence of ground states is determined by a threshold mass $\mu_p$, which itself is set by nonlocal Gagliardo–Nirenberg inequalities and marks the transition from nonexistence to existence as the mass crosses $\mu_p$ (Adami et al., 2018).
• Critical power in harmonic generation: In nonlinear optics, the threshold for temporal pulse collapse (“time-collapse critical power”) in third-harmonic generation is observed to be a well-defined value, independent of focusing conditions and lower than the fundamental wave’s critical power—a uniquely nonlocal critical-power phenomenon (Komatsubara et al., 2022).

These thresholds often correspond to bifurcations in the stationary or dynamical solution structure and serve as bifurcation points for switching, ejection, stability changes, or energy concentration.

4. Variational and Analytical Techniques in Nonlocal Critical Settings

Solving or analyzing nonlocal equations with critical-power nonlinearities requires specialized variational and analytical machinery:

• Nehari manifold and fiber map analysis: To locate multiple solutions, especially with sign-changing nonlinearities and critical growth, solutions are often sought as critical points constrained to the Nehari manifold, with further partition into subsets using fibering maps and analysis of second derivatives to distinguish minimizers/maximizers (1711.01854, Dhanya et al., 1 Mar 2025).
• Truncation and approximation arguments: For singular/critical problems where direct variational arguments are prevented by lack of differentiability or compactness, truncated or approximate problem sequences are constructed before passing to the limit (e.g., cut-off of bubbles/Talenti optimizers) (Mosconi et al., 2015, Ghanmi et al., 2022).
• Concentration-compactness and profile decomposition: To overcome the failure of the Palais–Smale condition at critical levels, advanced compactness decomposition is used to recover convergence after excluding mass loss or energy concentration—classically seen for (fractional) critical Sobolev exponents (Bhakta et al., 2020, Ó et al., 2015).
• Nonlocal energy functionals with critical exponents: The geometry of the energy landscape—often exhibiting mountain-pass or linking geometry, and sometimes requiring adaptation to non-smooth settings—plays a critical role in the existence and multiplicity of solutions to mixed local/nonlocal critical problems (Anthal et al., 2023, Ghanmi et al., 2022).
• Spectral and threshold estimates: In dynamical settings (e.g., damped wave or evolution equations), critical exponents affect decay estimates, global existence, and the sharpness of energy thresholds, with special attention to the role of spatial operators and kernel properties (Said, 2021, Zhao et al., 2021).

These methods are often intertwined, with the nonlocal and critical features of the operator and nonlinearity jointly dictating the functional framework and the valid range of analytical tools.

5. Physical and Applied Implications

Nonlocal critical-power nonlinearities have considerable impact in several application areas:

• Nonlinear optics: In media where the refractive index perturbation is determined by thermal or other nonlocal responses, the power threshold for soliton ejection or trapping is non-universal and tunable via defect strength and spatial configuration, enabling dynamic control of beam routing. For higher harmonics, distinct and lower “critical powers” govern pulse collapse, fundamental to filamentation dynamics and frequency conversion (Ye et al., 2010, Komatsubara et al., 2022).
• Disordered media and Anderson localization: Nonlocal nonlinearities increase the robustness of localized modes against destabilization, thus raising the threshold power for delocalization and modifying transport laws from exponential decay to power law (self-organized criticality) regimes (Folli et al., 2012, Iomin, 2019).
• Fractional and nonlocal PDEs: In mathematical physics, finance, and populations dynamics, equations with fractional Laplacians and critical nonlinearities model anomalous diffusion and critical transitions, with existence and uniqueness of solutions (and their multiplicity) depending on threshold parameters and domain/topology (Mosconi et al., 2015, Ghanmi et al., 2022).
• Nonlocal evolution and damped wave systems: The critical power threshold divides regimes of global existence from blow-up and determines the existence and structure of attractors; the damping and energy dissipation may be controlled by designing nonlocal weighting or dissipative terms (Said, 2021, Zhao et al., 2021).
• Mixed local–nonlocal systems: New classes of problems, where both local and fractional (nonlocal) diffusion act, show that critical exponents are dictated by the interplay of both operators. The combination of indefinite weight functions and critical growth yields bifurcations and multiplicity, controlled by the subcritical terms’ strength (Dhanya et al., 1 Mar 2025).

These phenomena underpin both the theoretical paper of nonlinear PDEs and the design of devices or protocols leveraging power-dependent switching, stability, and controlled transport.

6. Summary Table of Key Features and Models

Model/Setting	Operator(s)	Critical Exponent	Threshold/Bifurcation
Nonlocal NLS, thermal defect	Schrödinger + nonlocal response	Power: $U_{\max}$	Soliton ejection
Anderson localization, nonlocal NLSE	Fractional NLS, disorder	$P_c=\frac{|\beta_0|}{|\chi(0)|}$	Robustness to delocalization
Fractional (–Δ)^s, critical power	Fractional Laplacian	$2^*_s=\frac{2N}{N-2s}$	Existence/multiplicity
Hartree/Choquard critical term	Fractional Laplacian + HLS	$2^*_\mu$ from HLS	Mult. weak solutions
Mixed local/nonlocal operators	–Δp + (–Δq)^s	$p^*$, $q^*_s$	Dual bifurcation points
Damped σ-evolution, nonlocal NL	(–Δ)^σ + damping, Riesz pot.	$p_{crit}$ via α, m, σ	Global existence/blow-up
Third-harmonic pulse collapse	NL propagation, Kerr terms	Power: $P_{cr}^{(TH)}$	Temporal splitting

Each of these settings exemplifies the intricate and system-dependent character of nonlocal critical-power nonlinearities, where critical exponents, operator form, and domain/topological features jointly determine the qualitative dynamics of solutions.

7. Broader Impact and Directions

Nonlocal critical-power nonlinearities constitute a unifying framework for understanding threshold phenomena in complex media, incorporating the effects of extended interactions, power concentration, and the interplay between nonlocal operators and critical growth. The continuing development of analytical tools for such systems, especially in the presence of singularities, weights, or multi-scale effects, is fundamental for advancing both theoretical analysis and applied design in nonlinear science, from precise soliton control in optics to complex transport in disordered or anomalous media. Current research trajectories include extending results to variable-exponent frameworks, domains with more singular/heterogeneous structure, and the incorporation of stochastic effects or memory terms for generalized nonlocal models.