Upper Critical Fractional Choquard Equation

• The upper critical fractional Choquard equation is a nonlocal PDE defined by a critical convolution term per the Hardy–Littlewood–Sobolev inequality alongside singular local nonlinearities.
• It employs sophisticated variational frameworks in fractional Sobolev spaces, utilizing mountain-pass lemmas and concentration–compactness principles to address non-compactness issues.
• Research on this equation tackles analytical obstacles such as critical embeddings, weighted singular terms, and the existence, multiplicity, and stability of its solutions.

The upper critical fractional Choquard equation refers to nonlinear nonlocal PDEs where the convolution term reaches the critical scaling dictated by the Hardy–Littlewood–Sobolev (HLS) inequality and often coexists with local critical or singular nonlinearities. Such equations, formulated using the fractional Laplacian or fractional $p$-Laplacian, arise in mathematical physics, quantum mechanics, and nonlinear analysis. Their paper is distinguished by concentration-compactness phenomena, lack of compactness at critical exponents, and intricate variational structures incorporating both local and nonlocal effects.

1. Core Formulation and Critical Exponents

The prototypical upper critical fractional Choquard problem, as represented in bounded domains with possible singular coefficients, is

$$(-\Delta)_p^s u(x) = \lambda\, |u(x)|^{r-2}u(x)\,|x|^{-\alpha} + \gamma \bigg(\int_\Omega |u(y)|^q\, |x-y|^{-\mu}dy\bigg) |u(x)|^{q-2}u(x) \quad \text{in } \Omega,\qquad u=0\ \text{on } \mathbb{R}^N\setminus\Omega$$

with parameter constraints $p>1$, $0$N>sp$, $0\leq\alpha\leq sp$, $0<\mu<N$, $\lambda,\gamma>0$, $p\leq r\leq p^*_\alpha$, $p\leq 2q\leq 2p^*_{\mu,s}$, and critical exponents

$$p^*_\alpha = \frac{(N-\alpha)p}{N-sp},\qquad p^*_{\mu,s} = \frac{(N-\mu/2)\,p}{N-sp}.$$

The nonlocal Choquard term becomes upper-critical when $2q=2p^*_{\mu,s}$; the local term is "Hardy–Sobolev critical" at $r=p^*_\alpha$ (Yang et al., 2019).

In the classical Hilbert setting ($p=2$), upper-criticality typically refers to

$p = \frac{N+\alpha}{N-2s}$

so that the convolution

$\int_{\mathbb{R}^N}(I_\alpha*|u|^p)|u|^p$

scales as the fractional Sobolev norm (d'Avenia et al., 2014).

2. Variational Frameworks and Function Spaces

Solutions are sought in fractional Sobolev spaces (see $W_0^{s,p}(\Omega)$, the closure of $C^\infty_c(\Omega)$ under

$$[u]_{s,p}^p = \iint_{\mathbb{R}^N\times\mathbb{R}^N} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dx\,dy.$$

The energy functional becomes

$$J(u) = \frac{1}{p}[u]_{s,p}^p - \frac{\lambda}{r}\int_\Omega |u|^r|x|^{-\alpha}dx - \frac{\gamma}{2q}\iint_{\Omega\times\Omega} \frac{|u(x)|^q|u(y)|^q}{|x-y|^\mu}dx\,dy$$

with critical points corresponding to weak solutions.

Critical and upper-critical growth implies non-compactness of embeddings such as $W_0^{s,p}(\Omega)\hookrightarrow L^q(\Omega)$ at $q=p^*_{\mu,s}$, requiring refined methods for critical point identification (Yang et al., 2019, d'Avenia et al., 2014).

3. Compactness Breakdown and Analytical Obstacles

Compactness of Palais–Smale sequences is lost at upper criticality due to the scaling invariance of both the fractional Sobolev norm and the HLS convolution. Concentration and dichotomy (splitting) can occur. To recover compactness, one typically restricts attention to energy levels below sharp threshold constants—specifically,

$$c < \Bigl(\frac{p}{2p^*_{\mu,s}}\Bigr)\,S_{H,L}^{2p^*_{\mu,s}/(2p^*_{\mu,s}-p)}$$

where $S_{H,L}$ is the best constant associated to nonlocal critical embedding (Yang et al., 2019).

Regularity for weak solutions is ensured by fractional De Giorgi–Nash–Moser iterations and known boundary regularity estimates, with solutions typically lying in $C^{\beta}_{loc}(\Omega)$ for some $\beta\in(0,s)$ (Biswas et al., 2020, Mukherjee et al., 2016).

4. Existence, Multiplicity, and Structure of Solutions

Existence Theorems

In the upper-critical regime, under parameter constraints, there exists $\lambda^*>0$ such that for $\lambda\geq\lambda^*$, positive ground-state solutions exist in $W_0^{s,p}(\Omega)$ (Yang et al., 2019). Sign-changing least energy solutions are obtained by constrained minimization over decomposed Nehari manifolds.

Multiplicity

Multiplicity is established via category theory (Ljusternik–Schnirelmann) and genus arguments, relating the number of solutions to the topology of the set where the potential is minimized (Chen et al., 2019). For potentials $V(\varepsilon x)$ with multiple minima, solutions concentrate near these minima as $\varepsilon\to0$ (Aikyn et al., 30 Nov 2025).

Qualitative Properties

Solutions, when positive, enjoy regularity and radial symmetry (usually established via rearrangement arguments), and decay exponentially as $|x|\to\infty$ (Li, 2021, d'Avenia et al., 2014). The strong maximum principle ensures positivity for ground states. Nodal (sign-changing) solutions exist and can be characterized via Nehari manifold decompositions (Yang et al., 2019).

Stability and Instability

Ground states can be orbitally stable under suitable mass and energy constraints (Li, 2021), whereas mountain-pass solutions lying above certain thresholds can be strongly unstable (via virial criteria and blow-up arguments).

5. Singularities, Weighted Nonlinearities, and Hardy Terms

Inclusion of singular coefficients (weights such as $|x|^{-\alpha}$) and weighted Hardy terms requires characterizing best constants for Hardy–Sobolev and Stein–Weiss-type embeddings. Embedding results for weighted Morrey spaces and fractional Caffarelli–Kohn–Nirenberg inequalities are instrumental (Assunção et al., 2023).

In these settings, critical exponents adjust to include weights: $$p^*_{s}(\beta,\theta) = \frac{p(N-\beta)}{N-sp-\theta},\qquad p^{\sharp}_s(\delta,\theta,\mu) = \frac{p(N-\delta-\mu/2)}{N-sp-\theta}.$$ Mountain-pass geometry and new compactness criteria yield existence of weak solutions despite the singular and doubly critical structure.

6. Advanced Variational Techniques and Concentration–Compactness

The lack of compactness at upper-criticality mandates refined variational techniques:

• Mountain-pass Lemmas: Used to establish critical points below calculated thresholds, with explicit construction of paths.
• Truncation–penalization: A smooth cutoff freezes critical terms for large norms to enforce precompactness, particularly for mass-constrained problems (Aikyn et al., 30 Nov 2025).
• Concentration–compactness Principle: A generalized Lions lemma precludes vanishing and dichotomy via energy comparisons with bubble-type solutions (Mukherjee et al., 2016, Sakuma, 2023).
• Brezis–Lieb Splitting: Decomposes nonlocal potential terms under weak convergence, crucial for passing to the limit in critical regimes (Chen et al., 2019).
• Pohožaev Identities: These establish scaling relations and constrain critical points to proper manifolds, enabling uniqueness and exclusion of nonexistence in star-shaped domains (d'Avenia et al., 2014, Assunção et al., 2023).

7. Outlook and Open Problems

Open avenues include:

• Exploring orbital stability for normalized solutions at criticality (Aikyn et al., 30 Nov 2025).
• Extending theory to systems, mixed local-nonlocal nonlinearities, and even triply critical problems (Sakuma, 2023).
• Determining exact thresholds, bubble-extraction, and finer concentration behaviors as parameters approach critical regimes.
• Developing finer regularity, symmetry, and decay results for solutions in singular or weighted problem settings (Biswas et al., 2020, Assunção et al., 2023).

These analytical and variational developments profoundly impact fractional PDE theory, especially concerning nonlocal nonlinear Schrödinger equations, critical phenomena, and functional inequalities. The upper critical fractional Choquard equation remains a deeply influential subject, with its solution structure governed by variational geometry, embedding theorems, and concentration-compactness methodologies.