Choquard Nonlinearity at Critical Exponent

• The Choquard nonlinearity with upper critical exponent is a nonlocal equation coupling Riesz potential convolution with scaling invariance via the Hardy–Littlewood–Sobolev inequality.
• Variational methods incorporating the Pohožaev constraint and subcritical approximations enable recovery of groundstates despite critical loss of compactness.
• Analytical techniques such as concentration–compactness, fibering maps, and radial embeddings ensure existence, symmetry, and regularity of solutions.

The Choquard nonlinearity with upper critical exponent arises within a class of nonlocal (Hartree-type) nonlinear equations, where the convolution term features an exponent precisely determined by scaling invariance with respect to the underlying functional and the Hardy–Littlewood–Sobolev (HLS) inequality. This criticality presents variational and compactness challenges analogous to the local Sobolev embedding at critical exponent, but incorporates essential nonlocal features due to the convolution with the Riesz potential. The upper critical exponent demarcates the threshold beyond which standard compactness arguments fail, yet below it, groundstate and multiplicity results can often be recovered via sophisticated variational methods.

1. Formulation and the Upper Critical Hardy–Littlewood–Sobolev Exponent

Let $N \geq 3$ and $\alpha \in (0, N)$. The autonomous Choquard equation is given by

$$-\Delta u + u = (I_\alpha * |u|^p) |u|^{p-2}u \quad \text{in } \mathbb{R}^N,$$

where $I_\alpha(x) = \frac{A_\alpha(N)}{|x|^{N-\alpha}}$ is the Riesz potential, with $A_\alpha(N)$ the normalization constant. The HLS inequality ensures that the convolution is well-defined for $p$ within a precise range, but at

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

the problem becomes critical—a direct nonlocal analogue to the local Sobolev critical exponent $2^* = \frac{2N}{N-2}$ (Li et al., 2018). In bounded domains or with fractional Laplacian, the critical exponent structure adapts, e.g., $2^*_{\mu,s} = \frac{2n-\mu}{n-2s}$ (Mukherjee et al., 2016).

The nonlocal term's criticality means the associated energy functional is often only weakly lower semicontinuous, and standard minimization may fail due to loss of compactness caused by bubble-type solutions and translation invariance.

2. Variational Framework and Pohožaev Constraint Methods

The natural energy associated to the Choquard equation is

$$E(u) = \frac{1}{2} \int_{\mathbb{R}^N} (|\nabla u|^2 + u^2) \, dx - \frac{1}{2p} \int_{\mathbb{R}^N} (I_\alpha * |u|^p) |u|^p\,dx.$$

Critical points of $E: H^1(\mathbb{R}^N) \rightarrow \mathbb{R}$ correspond to weak solutions. At the upper critical exponent, the Pohožaev identity

$$\frac{N-2}{2}\int |\nabla u|^2\,dx + \frac{N}{2}\int |u|^2\,dx = \frac{N+\alpha}{2p^*}\int (I_\alpha*|u|^{p^*})|u|^{p^*}\,dx$$

defines a natural constraint $P(u) = 0$, where $P$ is the Pohožaev functional. The set $$\mathcal{M} = \{u \in H^1 \setminus\{0\} : P(u) = 0\}$$ (the "Pohožaev manifold") becomes the minimization space for groundstates (Li et al., 2018, Li et al., 2018). Minimizers on $\mathcal{M}$, via Lagrange multiplier theory and scaling arguments, yield groundstates that satisfy both the Euler–Lagrange equation and the Pohožaev constraint.

3. Subcritical Approximation and Compactness Recovery

Due to loss of compactness at $p^*$, existence is often proved by a subcritical approximation. For $p<p^*$, the Sobolev embeddings are compact (particularly in radial symmetry), and groundstate minimizers $u_p$ exist. By constructing a family $p_n \nearrow p^*$ and establishing uniform $H^1$ bounds, weak limits $u_p \rightharpoonup u^*$ are extracted. Energy and constraint estimates show $u^*\not\equiv 0$ and the limiting solution is a groundstate for the critical problem (Li et al., 2018, Li et al., 2018).

Strauss's radial compactness lemma is central: for bounded radial sequences in $H^1(\mathbb{R}^N)$, strong convergence in all $L^r$ spaces below the Sobolev critical exponent is guaranteed, permitting passage to the limit in nonlocal terms.

4. Existence, Symmetry, and Regularity of Groundstates

For the autonomous Choquard equation at $p^*$, one obtains a positive, radially symmetric groundstate $u_{p^*}\in H^1_{\mathrm{rad}}(\mathbb{R}^N)$:

$$-\Delta u + u = (I_\alpha * |u|^{p^*}) |u|^{p^* - 2} u, \quad u > 0.$$

The groundstate minimizes the constrained energy on the Pohožaev manifold, exploits compactness via radial embeddings, and, by elliptic regularity, achieves $u\in C^{1,\beta}$ for all $\beta<1$. The strong maximum principle secures strict positivity (Li et al., 2018, Li et al., 2018).

5. Critical Phenomena: Loss of Compactness, Concentration–Compactness, and Multiplicity

At $p^*$, minimizing sequences can lose mass to infinity, or concentrate to bubbles—the limiting HLS optimizers. Concentration–compactness and nonlocal Brezis–Lieb-type lemmas control this behavior, precluding vanishing or dichotomy when energy stays below a computable threshold. In bounded domains, the lack of compactness leads to the necessity for topological or critical-point-at-infinity arguments; e.g., the presence of nontrivial domain topology can enforce existence via homological methods (Alghamdi et al., 2024, Chtioui et al., 25 Jan 2026).

In strongly indefinite settings or with external potentials, linking structures and spectral splits replace mountain-pass geometry, with sharp threshold values governed by best HLS constants (Gao et al., 2017, Mukherjee et al., 2016).

6. Extensions: Fractional and Kirchhoff Operators, Variable Exponents, and Exponential Criticality

Generalizations include Choquard equations with a fractional Laplacian (Mukherjee et al., 2016, Goel et al., 9 Sep 2025), Kirchhoff-type nonlocal operators (Shang et al., 18 Sep 2025, Goel et al., 30 Aug 2025), and critical exponential nonlinearities in dimensions or function spaces where power-type criticality is replaced by Moser–Trudinger-type critical growth (Kanungo et al., 2024, Gao et al., 4 Aug 2025, Böer et al., 2020, Romani, 2024).

In variable exponent frameworks, the critical exponent $p_s^*(x)$ depends on location, requiring refined compact embedding results even at criticality (Sakuma, 2024). The passage to criticality in these contexts leverages log-Hölder conditions and sharp "touching rate" hypotheses, enabling recovery of the Palais–Smale condition.

7. Analytical Techniques and Impact of Problem Data

Techniques include:

• Use of sharp HLS inequalities for convolution estimates.
• Fibering map/scaling analysis for energy and constraint maximization.
• Topological and variational tools: Mountain Pass Theorem, Ekeland’s principle, genus and linking theory, concentration–compactness.
• Critical-point-at-infinity and bubble analysis for failure of Palais–Smale.
• Pohožaev constraint minimization to bypass direct compactness.
• Regularity theory: bootstrapping from weak solutions to $C^{1,\beta}$ functions.

The effect of boundary geometry and external potential influences existence and multiplicity, with convexity of the Neumann boundary in mixed boundary problems lowering energy quotients and enabling groundstates (Chtioui et al., 25 Jan 2026).

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The Choquard upper critical exponent regime is typified by sharp variational structures, critical loss of compactness, and rich geometric and analytical phenomena, with existence, regularity, and multiplicity hinging on delicate estimates rooted in the HLS inequality and Pohožaev-type arguments.