Local Pohozaev Identity in Elliptic PDEs

• Local Pohozaev identity is a functional analytic tool that relates volume integrals of elliptic PDE solutions with local boundary integrals reflecting domain geometry.
• It extends classical identities to fractional, anisotropic, and nonlocal operator settings using scaling arguments, nonlocal integration by parts, and boundary regularity analysis.
• The identity provides critical insights for proving nonexistence, unique continuation, and blow-up analysis in various nonlinear and geometric PDE problems.

The local Pohozaev identity is a functional analytic tool that yields explicit relations between integrals of solutions to elliptic partial differential equations and certain boundary integrals, encapsulating the interplay between the equation, its nonlinearity, and the geometry of the domain. While the classical Pohozaev identity applies to second-order differential operators, recent advances have generalized its local form to fractional, nonlocal, and anisotropic settings, revealing deeper geometric structures and criticality phenomena.

1. Canonical Formulation for the Fractional Laplacian

Given the semilinear Dirichlet problem for the fractional Laplacian: $$\begin{cases} (-\Delta)^s u = f(u) & \text{in } \Omega, \ u \equiv 0 & \text{in } \mathbb{R}^n \setminus \Omega, \end{cases}$$ where $s\in(0,1)$, $\Omega\subset\mathbb{R}^n$ is a bounded $C^{1,1}$ domain, and $\delta(x) = \operatorname{dist}(x,\partial\Omega)$, the local Pohozaev identity reads: $$\boxed{ (2s-n)\int_\Omega u f(u) dx + 2n\int_\Omega F(u) dx = \Gamma(1+s)^2 \int_{\partial\Omega} \left( \frac{u}{\delta^s} \right )^2 (x \cdot \nu) d\sigma }$$ with $F(u) = \int_0^u f(\tau) d\tau$, and $\nu$ the exterior unit normal. The boundary term is local and depends only on the trace of $u/\delta^s$ at $\partial \Omega$, which replaces the classical normal derivative $\partial_\nu u$ in Pohozaev's original formula (Ros-Oton et al., 2012).

2. Generalizations to Higher-Order, Anisotropic, and Nonlocal Operators

Local Pohozaev identities have been established for a variety of operators beyond the fractional Laplacian, including:

• Anisotropic, Lévy-type and variable-coefficient fractional operators: For a symmetric, $2s$-order integro-differential operator $L$ (with spectral measure $a$), the Pohozaev identity takes the form:

$$\int_{\Omega} (x\cdot\nabla u) Lu\,dx = \frac{2s-n}{2}\int_\Omega u Lu\,dx - \frac{\Gamma(1+s)^2}{2}\int_{\partial\Omega} \mathcal{A}(\nu)\left(\frac{u}{d^s}\right)^2 (x\cdot\nu) d\sigma$$

where $d(x)$ is distance to the boundary, and $\mathcal{A}(\nu)$ aggregates anisotropy effects via the spectral measure (Ros-Oton et al., 2015).

• Variable-coefficient pseudodifferential operators: For general elliptic classical pseudodifferential operators $P$ of order $2a$ with even symbol and satisfying transmission conditions, the local Pohozaev identity explicitly involves the principal symbol evaluated at the boundary normal and boundary traces $d^{-a}u|_{\partial\Omega}$ (Grubb, 2015).
• Higher order fractional Laplacians ($s>1$): The identity structure remains, with the continuous boundary trace $\frac{u}{d^s}$, replacing higher order derivatives (Ros-Oton et al., 2014).
• Grushin-type degenerate sub-Laplacians and anisotropic $p$-Laplace-type equations: Pohozaev identities are obtained via domain variation, providing local (translating and scaling) identities in sub-domains, with explicit degeneracy-related terms reflecting the operator's non-uniform ellipticity (Wei et al., 26 Jul 2025, Anthal et al., 10 Jun 2025).

3. Analytical Framework and Boundary Regularity

Establishing the local Pohozaev identity in fractional and nonlocal settings demands careful analysis of boundary regularity. For bounded solutions:

• $u \in C^s(\mathbb{R}^n)$,
• $u/\delta^s \in C^\alpha(\overline \Omega)$ for some $\alpha > 0$,
• Near $\partial \Omega$, $|\nabla u| \lesssim \delta^{s-1}$.

This ensures traces like $u/\delta^s|_{\partial\Omega}$ (and analogs for other operators) are well-defined and admit the necessary boundary regularity to interpret the integral correctly (Ros-Oton et al., 2012, Ros-Oton et al., 2015).

4. Methodology: Nonlocal Integration by Parts and Scaling Arguments

The derivation of the local Pohozaev identity typically unfolds through:

• Scaling-in-λ arguments: Analysis of $\int_{\Omega} u_\lambda(x) L u(x)\,dx$ under dilation produces both volume and boundary terms.
• Nonlocal integration by parts: Plancherel-type formulas, Fourier-based methods, and factorization of operators (e.g., $P \sim P^-P^+$) yield explicit boundary bilinear forms in terms of transmission traces (Ros-Oton et al., 2012, Grubb, 2015).
• Boundary singularity analysis: One-dimensional reductions capture singular boundary behavior, extracting the boundary coefficient (often involving gamma functions and geometric quantities).

5. Comparison with Classical Pohozaev Identity

In the classical setting for $-\Delta u = f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, the identity is: $$(2-n)\int_\Omega |\nabla u|^2 dx + 2n \int_\Omega F(u)\,dx = \int_{\partial\Omega} (x\cdot\nu) (\partial_\nu u)^2 d\sigma$$ For nonlocal/fractional problems, the "normal derivative" is systematically replaced by $u/\delta^s|_{\partial\Omega}$ or its appropriate generalization (Ros-Oton et al., 2012, Ros-Oton et al., 2015, Ros-Oton et al., 2014, Biswas, 2024). The boundary term remains local despite the nonlocal nature of the interior operator.

6. Applications: Nonexistence, Unique Continuation, and Blow-Up Analysis

The local Pohozaev identity underpins several foundational results:

• Nonexistence in star-shaped domains: For nonlinearities exceeding the critical Sobolev exponent ($p \geq \frac{n+2s}{n-2s}$), no nontrivial bounded solution exists (Ros-Oton et al., 2012, Ros-Oton et al., 2015, Ros-Oton et al., 2014).
• Unique continuation: If the trace $u/\delta^s$ vanishes on the boundary, the solution must vanish in the whole domain (Ros-Oton et al., 2014, Ros-Oton et al., 2015).
• Blow-up and bubble analysis: The local identity quantifies concentration phenomena and pinpoints locations of bubbles in critical elliptic systems via reduction techniques (Guo et al., 2024, Guo et al., 2019).

Applications to control theory, especially for singular Schrödinger operators, leverage local Pohozaev identities to study boundary controllability and derive critical energy manifolds (Cazacu, 2011).

7. Extensions: Systems, Removability of Singularities, and Geometric Interpretations

• Systems of equations: The local Pohozaev identity extends to elliptic systems with coupled nonlinearities, encoding synchronized concentration and symmetry conditions (Guo et al., 2024, Biswas, 2024).
• Removability of singularities: For super-Liouville type equations on Riemann surfaces, the vanishing of the Pohozaev "constant" is both necessary and sufficient for removability at isolated singularities. This marks a departure from reliance on conformal invariance, situating the Pohozaev identity as a universal criterion for singularity analysis (Jost et al., 2017).
• Geometric structure: The identity reveals deep links between critical exponents, geometry (star-shapedness, conical singularities), and analytic properties of solutions; the boundary flux measures weighted geometric distributions.

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The local Pohozaev identity thus generalizes the classical energy–flux balance to broad classes of elliptic and nonlocal equations, serving as a central tool in nonlinear analysis, spectral theory, and geometric PDEs. Its scope, through rigorous boundary regularity and precise explicit formulae, is foundational for nonexistence results, unique continuation, criticality phenomena, and geometric quantization (Ros-Oton et al., 2012, Ros-Oton et al., 2015, Ros-Oton et al., 2014, Grubb, 2015, Biswas, 2024, Jost et al., 2017).