Local Pohozaev Identities in PDE Analysis

• Local Pohozaev identities are sharp integral relations obtained by multiplying a PDE by a geometrically motivated test function and integrating by parts.
• They are vital for ruling out existence, isolating blow-up profiles, and encoding conservation laws in elliptic, nonlocal, and quasilinear PDE contexts.
• Recent extensions apply these identities to anisotropic and mixed operators, enhancing finite-dimensional reduction techniques and concentration analysis.

A local Pohozaev identity is a sharp integral relation, typically derived by multiplying a partial differential equation (PDE) by a geometrically-motivated test function (often, a component of the underlying infinitesimal symmetry group, such as scaling or translation), and integrating by parts over a bounded subdomain. These identities provide pointwise constraints or balance laws linking the behavior of solutions in the interior of a domain to their boundary traces. In the classical setting, the Pohozaev identity is a decisive tool for rule out existence, isolate blow-up profiles, or encode hidden conservation laws in elliptic and geometric PDEs. In modern theory, 'local Pohozaev identities' have emerged as essential in problems with nonlocal, anisotropic, mixed or quasilinear operators, and in the analysis of complex concentration or blow-up phenomena.

1. General Structure and Formulation

The fundamental paradigm is to consider an elliptic PDE or system (possibly local, nonlocal, or quasilinear): $\mathcal{L} u = f(u) \quad \text{in } \Omega,$ subject to Dirichlet-type boundary conditions. Here, $\mathcal{L}$ may denote a classical Laplacian, a fractional Laplacian $(-\Delta)^s$, an anisotropic or space-dependent integro-differential operator, a (fractional) $p$-Laplacian, or a combination thereof.

The prototypical local Pohozaev identity, for scalar semilinear problems, is: $$\int_{\partial D} \Big[ \frac{\partial u}{\partial \nu} (x \cdot \nabla u) - \frac{1}{2} |\nabla u|^2(x \cdot \nu) \Big]\, dS = \int_D \left( \frac{n-2}{2} f(u) u - n F(u) \right) dx,$$ where $F(u)=\int_0^u f(s) ds$, $D\subset\Omega$ is a local subdomain, and $\nu$ is the unit outer normal. Extensions to nonlocal or quasilinear settings involve highly nontrivial constructions: e.g., for the fractional Laplacian one obtains (in the notation of (Ros-Oton et al., 2012, Ros-Oton et al., 2014, Ros-Oton, 2017)): $$(2s-n)\int_D u f(u) dx + 2n \int_D F(u) dx = \mathcal{B}(u)[\partial D],$$ where the boundary term $\mathcal{B}(u)[\partial D]$ depends locally on the quotient $u/\delta^s$ (with $\delta(x)=\mathrm{dist}(x,\partial D)$), and in the most general setting, may involve additional weight functions reflecting operator anisotropy or order.

The structure generalizes to systems, mixed local-nonlocal problems, equations with critical nonlocal convolution nonlinearities (as in Choquard–Hartree-type equations (Gao et al., 2022, Cassani et al., 7 Feb 2024)), degenerate or subelliptic operators (e.g., of Grushin-type (Wei et al., 18 Apr 2024, Wei et al., 26 Jul 2025)), or to locally conserved geometric quantities on manifolds with boundary (Gover et al., 2010).

2. Origin and Derivation: Symmetry, Conservation, and Integration by Parts

The derivation of a local Pohozaev identity is intrinsically tied to infinitesimal symmetries, variational structure, and the ability to perform integration by parts or an appropriate nonlocal substitute. Key ingredients include:

• Infinitesimal Conformal Symmetry: In geometric settings, conformal Killing vector fields $X$ (those satisfying $\mathcal{L}_X g = 2(\mathrm{div}\, X)\, g$ for the metric $g$) single out natural variations under which functionals are invariant. The unified Pohozaev–Schoen identity on manifolds with boundary stems precisely from such symmetry (Gover et al., 2010):

$$\int_M (\mathcal{L}_X V) \, d\mathrm{vol} = -n \int_{\partial M} B^\circ(X,v) d\sigma,$$

with $B$ a locally conserved symmetric $2$-tensor, $V$ its trace, and $B^\circ$ its trace-free part.

• Nonlocal Integration by Parts: In fractional and nonlocal settings, classical divergence computations are replaced by delicate bilinear forms, kernel-weighted double integrals, or extension methods (e.g., Caffarelli–Silvestre extension). For example, using rescalings $u_\lambda(x)=u(\lambda x)$ leads to identities (see (Ros-Oton et al., 2012, Ros-Oton et al., 2014, Ros-Oton, 2017)):

$$\int_{\Omega} (x \cdot \nabla u) \, \mathcal{L} u\, dx = \text{weighted bulk terms} - \text{(weighted) boundary contribution}.$$

For regional operators, integration by parts must incorporate explicit remainder terms accounting for the restriction to $\Omega$ (Djitte, 29 Jul 2025).

• Commutator and Symbol Analysis: For $x$-dependent or general pseudodifferential operators, commutator estimates and symbol calculus ensure that lower-order perturbations or nonsymmetric parts generate controllable (often vanishing) error terms in the local formula (Grubb, 2015).
• Domain Variation and Local Testing: When the global identity is insufficient (as is common in blow-up, concentration, or critical exponent problems), 'localization' via characteristic functions or smooth cutoffs around a candidate point yields a family of local identities that capture the transfer of energy or mass across spatial scales (Bartolucci et al., 2019Grossi et al., 2021Guo et al., 26 Sep 2024).

3. Key Features in Modern Applications

3.1. Nonlocality and Boundary Regularity

In nonlocal problems (fractional Laplacian, anisotropic integro-differential operators), although the operator itself is globally defined, the Pohozaev identity often isolates local boundary traces. Specifically, even for the nonlocal operator $(-\Delta)^s$ in a bounded domain with zero Dirichlet data, the identity reduces to a boundary integral involving $(u/\delta^s)^2$ (see (Ros-Oton et al., 2012, Ros-Oton et al., 2014, Ros-Oton et al., 2015)): $$\mathcal{B}(u)[\partial \Omega] = \Gamma(1+s)^2 \int_{\partial \Omega} \left(\frac{u}{\delta^s}\right)^2 (x \cdot \nu) d\sigma.$$ This emergence of a completely local boundary term is both surprising and crucial: it allows for direct analogues of classical uniqueness, nonexistence, and rigidity phenomena.

The boundary regularity of $u/\delta^s$ (or its analogues in the anisotropic/mixed setting) becomes a central technical point. Regularity up to the boundary, sometimes only up to H\"older continuity, is essential to justify passing from bulk to boundary in the derivation.

3.2. Local Identities and Finite Dimensional Reduction

Local Pohozaev identities are an indispensable ingredient in finite-dimensional reduction and concentration compactness approaches to critical and supercritical nonlocal problems. By restricting the identity to small neighborhoods of peak points and incorporating the approximate blow-up profiles (usually, Talenti or standard bubble solutions), one obtains algebraic constraints ('reduced systems') for parameters locating the spikes or bubbles (Guo et al., 2019Gao et al., 2022Cassani et al., 7 Feb 2024Guo et al., 26 Sep 2024).

For vector-valued, coupled, or critical elliptic systems, these local identities allow one to transcend the limitations posed by weak symmetry, degenerate coupling exponents, or lack of direct access to derivatives of the reduced energy functional (see (Guo et al., 26 Sep 2024)).

3.3. Extensions to Geometric and Physical Contexts

In geometric analysis, local Pohozaev (Kazdan–Warner–Schoen) identities provide necessary constraints for the scalar curvature prescription problem, Q-curvature, and the understanding of conformal invariants in dimensions $n \ge 3$ (Gover et al., 2010). In physics, the complementarity between Pohozaev-type identities and conservation laws, particularly Noether/Hilbert currents, elucidates the balance laws for variational models in general relativity and field theories.

In the investigation of singularities (particularly, removable or essential singularities of solutions), the validity of a local Pohozaev identity (often encoded as the vanishing of a suitably defined 'Pohozaev constant') is shown to be a necessary and sufficient condition for removability (Jost et al., 2017). When conformal invariance breaks down, as in problems with conical singularities, the Pohozaev constant measures the local defect.

4. Unique Continuation, Nonexistence, and Spectral Applications

The presence of a sharp local Pohozaev identity underpins powerful qualitative results:

• Unique Continuation: If the boundary (or "fractional normal derivative") trace of an eigenfunction vanishes (e.g., $u/\delta^s|_{\partial \Omega} = 0$), then $u$ must vanish identically inside ($u \equiv 0$). This boundary unique continuation is nontrivial for nonlocal problems (Ros-Oton et al., 2014, Ros-Oton, 2017, Grubb, 2015, Djitte, 29 Jul 2025, Biswas, 22 Oct 2024).
• Nonexistence for Supercritical Problems: If the nonlinearity is above the critical threshold, local Pohozaev identities can show that, in star-shaped (or suitably symmetric) domains, any nontrivial solution would generate a sign-contradiction in the boundary integral, implying nonexistence (1207.59861502.01431Ros-Oton, 2017Grubb, 2015Biswas, 22 Oct 2024Anthal et al., 10 Jun 2025).
• Spectral Theory: In nonlocal eigenvalue problems, the identity can enforce simplicity of the spectrum or exclude incompatible mass concentration, guiding the shape optimization or control theory for fractional operators (1406.11071705.05525).
• Characterization of Blow-Up and Concentration: In applications to critical equations with multiple bumps or bubbles (e.g., (Guo et al., 2019Gao et al., 2022Cassani et al., 7 Feb 2024Guo et al., 26 Sep 2024)), the local identities determine the permissible locations, rates, and topological constraints for concentration phenomena.

5. Examples and Explicit Formulas

A non-exhaustive assortment of explicit local Pohozaev identity forms drawn from the literature includes:

Operator/context	Identity Summary	Reference
Local Laplacian, semilinear	$$\int_{\partial D}[\dots] = \int_D \left( \frac{n-2}{2} f(u)u - nF(u) \right) dx$$	(Ros-Oton, 2017)
Fractional Laplacian	$$(2s-n)\int_D u f(u)dx + 2n\int_D F(u)dx = \Gamma(1+s)^2\int_{\partial D} (u/\delta^s)^2(x\cdot\nu)d\sigma$$	(Ros-Oton et al., 2012)
Higher order fractional	Similar as above, constants via $\Gamma(1+s)^2$; boundary term with $u/d^s$	(Ros-Oton et al., 2014)
Anisotropic integro-differential	$$(2s-n)/2 \int_{D} uL u dx - c\int_{\partial D} \mathcal{A}(\nu)(u/d^s)^2(x\cdot\nu)d\sigma$$	(Ros-Oton et al., 2015)
Regional fractional	$$-\int_{\Omega} \delta^{2-2s}(x)x\cdot\nabla u(x)(-\Delta)^s_\Omega u(x)dx = \frac{N-2s}{2}[u]^2_{H^s(\Omega;\delta^{2-2s})} - d_s \int_{\partial \Omega} (u/\delta^{2s-1})^2x\cdot\nu d\sigma + \mathcal{R}[u]$$	(Djitte, 29 Jul 2025)
Quasilinear local and nonlocal (p-Laplace)	$$\alpha\frac{n-p}{p} \|\nabla u\|^p + \beta\frac{n-sp}{p}[u]^p_{s,p} = n\int_{\mathbb{R}^n} F(u)dx$$	(Anthal et al., 10 Jun 2025)
Elliptic system via finite reduction	$$\int_{D_\rho} \partial_{y_i}P(y)u^2 + \partial_{y_i}Q(y)v^2\,dy = o(1)$$	(Guo et al., 26 Sep 2024)

This table illustrates the flexibility of the Pohozaev identity template, accommodating the subtleties introduced by operator order, anisotropy, nonlocality, and system structure.

6. Impact and Contemporary Scope

Local Pohozaev identities now form the backbone of critical point theory, blow-up analysis, and concentration compactness arguments in nonlocal and geometric PDE contexts. They are vital in:

• Establishing energy quantization and energy balance in limiting processes,
• Classifying and ruling out nontrivial or sign-changing solutions in regimes where compactness is lost,
• Capturing boundary phenomena for spectral analysis or control problems,
• Providing analytical characterization of the removability (or persistence) of singularities in nonlinear elliptic equations and geometric problems,
• Transferring local geometric or topological features of the domain or the coefficients (e.g., critical points of $r^2V(r,x'')$ in multi-bubble problems).

The broad applicability spans classical and fractional PDEs, nonlinear systems, geometric analysis, and mathematical physics, with increasing relevance to emerging areas such as regional fractional problems, degenerate subelliptic equations, and problems with mixed local–nonlocal structure.

7. Outlook and Ongoing Developments

Recent advances point toward further generalization of local Pohozaev identities, including:

• Quasilinear and anisotropic operators with mixed boundary conditions,
• Doubly nonlocal systems (coupling fractional and convolution operators),
• Degenerate or hypoelliptic structures (Grushin, sub-Laplacian),
• Explicit defect/correction terms (remainder terms) for Dirichlet and regional operators (Djitte, 29 Jul 2025),
• Identification of sharp removable singularity criteria via a 'Pohozaev constant' (Jost et al., 2017),
• Application to uniqueness, nondegeneracy, and symmetry-breaking analysis in multi-component and critical exponent systems (Guo et al., 26 Sep 2024).

The methodological synthesis of symmetry, variational calculus, operator theory, and asymptotic analysis underlying local Pohozaev identities ensures their centrality in both deep theoretical questions and concrete applied analysis of PDEs.