Pohozaev Identity in PDE Analysis

• Pohozaev type identity is a collection of integral identities that capture conserved or balance relations for regular solutions of variational PDEs and geometric problems.
• It is derived via methods like integration by parts with scaling vector fields, extending to nonlocal, fractional, and anisotropic operators.
• These identities have practical applications in nonexistence proofs, spectral theory, and geometric rigidity, influencing modern approaches in PDE analysis.

A Pohozaev type identity is a family of integral identities providing conserved or balance relations that must be satisfied by sufficiently regular solutions to variational partial differential equations or geometric variational problems. Such identities reflect hidden symmetries—commonly conformal or scaling invariance—and have become a central analytic tool in the paper of PDEs, geometry, and mathematical physics, with deep implications for nonexistence, uniqueness, regularity, and qualitative structure of solutions.

1. Classical and Geometric Origins

The classical Pohozaev identity was introduced in the context of semilinear elliptic equations on bounded domains with boundary, such as

$$\Delta u + f(u) = 0 \quad \text{in}\quad \Omega, \;\; u = 0 \;\; \text{on}\; \partial\Omega.$$

For $u$ sufficiently regular and $\Omega \subset \mathbb{R}^n$ smooth, one multiplies the equation by the scaling (Euler) vector field $x\cdot\nabla u$ and integrates by parts, yielding

$$\frac{n-2}{2}\int_{\Omega} f(u) u\,dx - n\int_{\Omega} F(u)\,dx = \frac{1}{2}\int_{\partial\Omega} (x\cdot\nu) \left|\frac{\partial u}{\partial\nu}\right|^2 d\sigma,$$

where $F$ is an antiderivative of $f$, and $\nu$ is the unit outward normal. The identity constrains possible solutions, with important nonexistence results if $(x\cdot\nu)\geq 0$ (e.g., star-shaped domains) and the energy contributions are sign-definite (Gover et al., 2010).

The geometric generalization appears in the context of conformal geometry. For conformally variational natural scalar invariants (e.g., the scalar curvature, $Q$-curvature), and more generally traces of locally conserved symmetric 2-tensors, the identity extends to any Riemannian manifold $(M,g)$ with conformal vector field $X$: $$\int_M \mathcal{L}_X V\,d\mathrm{vol}_g = -\frac{2}{n}\int_{\partial M} B_0(X,\nu)\,d\sigma,$$ where $V = \mathrm{tr}_g B$, $B_0$ is the trace-free part of $B$, and $\mathcal{L}_X$ is the Lie derivative along $X$. When $M$ is closed, the right-hand side vanishes, giving a Kazdan–Warner type identity (Gover et al., 2010).

Schoen's unifying principle (notably for $n \geq 3$) captures both the Pohozaev and Kazdan–Warner identities and arises from the gauge invariance of conformally invariant functionals. The underlying variational structure and Noetherian conservation laws ensure that solutions constrained by symmetry must satisfy these identities.

2. Nonlocal and Fractional Operators

A key advance is the extension of Pohozaev-type identities to fractional and more general nonlocal operators. For bounded solutions $u$ of the semilinear problem with the fractional Laplacian,

$$(-\Delta)^s u = f(u) \;\;\text{in}\; \Omega\, (\text{with}\; s\in(0,1),\; u = 0 \;\;\text{in}\; \mathbb{R}^n \setminus \Omega),$$

one obtains the identity (Ros-Oton et al., 2012, Ros-Oton et al., 2012),

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

where $\delta(x)=\operatorname{dist}(x,\partial\Omega)$. The nonlocality is encoded in the term $u/\delta^s$ on the boundary, which generalizes the normal derivative for $s=1$.

For translation-invariant, variable-coefficient, or higher-order fractional pseudodifferential operators $P$ of order $2a$, the identity generalizes to (Grubb, 2015)

$$\int_\Omega [P u (x\cdot\nabla u') + (x\cdot\nabla u) P^* u'] dx = \text{(explicit boundary term)} + \int_\Omega [P, x\cdot V] u\,u'\,dx,$$

where the commutator $[P, x\cdot V]$ corrects for loss of translation invariance, and the boundary term depends on the trace of $d^{-a}u|_{\partial\Omega}$ (with $d(x)=\operatorname{dist}(x,\partial\Omega)$).

For the fractional $p$-Laplacian on $\mathbb{R}^n$,

$(-\Delta)_p^s u = g(u)$

the identity reads (Ambrosio, 2023): $$C_{N,s,p}(N-sp) \iint_{\mathbb{R}^{2n}} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\,dx\,dy - N\int_{\mathbb{R}^n} G(u)\,dx = 0,$$ where $G(u) = \int_0^u g(t)\,dt$.

3. Anisotropic, Quasilinear, and Degenerate Operators

Pohozaev-type identities have been extended to anisotropic and degenerate frameworks. For quasilinear Finsler anisotropic equations,

$$-\operatorname{div}(B'(H(\nabla u))\nabla H(\nabla u))=g(x, u)$$

with $B$ and $H$ satisfying suitable homogeneity and ellipticity, the identity becomes (Montoro et al., 2022): $$N\int_{\Omega} G(x, u)\,dx + \int_{\Omega} (x\cdot\nabla_x G(x, u))\,dx - N\int_{\Omega} B(H(\nabla u))\,dx + \int_{\Omega} B'(H(\nabla u))H(\nabla u)\,dx = \text{boundary terms},$$ with $G(x, u) = \int_0^u g(x,s)\,ds$.

For equations involving Hörmander vector fields (subelliptic settings), e.g.,

$$\operatorname{div}_x(F_p(x,u,V_X u)) + F_z(x, u, V_X u) = 0,$$

where $V_X u$ is the $X$-gradient for a family of homogeneous vector fields, a general star-shaped Pohozaev-type identity relates the energy, anisotropic homogeneity, and nonlinearities to boundary terms constructed with the infinitesimal generator of the non-isotropic dilations (Biagi et al., 2020).

For degenerate or singular weights as in

$$-\operatorname{div}(t^{1-2s}A(x,t)\nabla U) + t^{1-2s}c(x,t) = 0,$$

and nonhomogeneous Neumann data, a Pohozaev-type identity can be rigorously established using fine Sobolev estimates for weighted spaces, enabling transfer of interior regularity to precisely defined boundary integrals (Felli et al., 2022).

4. Application to Nonexistence, Multiplicity, and Geometric Rigidity

Pohozaev-type identities have concrete applications:

• Nonexistence: When the boundary integrals have a controlled sign (for instance, $x\cdot\nu\geq 0$ on $\partial\Omega$), identities may force certain bulk integrals to vanish, precluding nontrivial solutions to associated PDEs with supercritical nonlinearities (Ros-Oton et al., 2012, Anthal et al., 10 Jun 2025).
• Spectral theory: In the paper of fractional Laplacian and eigenvalues, generalized Pohozaev identities have been used to derive Hadamard formulas for eigenvalue variation under domain deformations and to establish the simplicity of radial eigenvalues (Djitte et al., 2021).
• Multiplicity and signatures: Some identities remain valid for all growth regimes of the nonlinearity (including supercritical), hence can be used to count or exclude multiple solutions in degenerate/anisotropic equations and problems involving the $1$-Laplacian (Molino et al., 2017).
• Geometry and rigidity: In Riemannian geometry, Pohozaev–Schoen-type identities imply rigidity statements for Ricci–solitons and establish almost–Schur inequalities on asymptotically Euclidean manifolds, providing criteria for flatness or uniqueness in geometric flows (Freitas et al., 2020).

5. Conformal and Variational Structure

The unifying mechanism is the interaction between infinitesimal symmetries (conformal, scaling, Möbius) and variational structure. In particular:

• Noether's Theorem and Conservation Laws: In Lagrangian or Hamiltonian systems, energy-momentum tensors associated to underlying symmetries are divergence-free, and their contraction with symmetry-generating vector fields produces integral identities. This framework neatly explains the appearance of Pohozaev-type balances in geometric PDEs and physics (Gover et al., 2010, Bozhkov et al., 2011).
• Extensions to Boundary Singularities: For singular potentials (e.g., Schrödinger operators with Hardy-type singularity at the boundary), Pohozaev identities must be reformulated in natural weighted Sobolev spaces. The boundary integrals require refined trace regularity to be defined, as in the case of $$\int_{\partial\Omega} (x\cdot\nu)(\partial u/\partial\nu)^2 d\sigma$$ with $|x|$-weights (Cazacu, 2011).

6. Fractional Green Function and Robin Function Identities

For the fractional Laplacian, there exists a Pohozaev-type identity connecting the Green function $G_s(x,z)$ (with regular part $H_s$) and Robin function $\mathcal{R}_s(x)$. The main bilinear identity is (Dieb et al., 3 Jun 2025): $$\Gamma^2(1+s)\!\! \int_{\partial\Omega}\! \frac{G_s(x,z)}{\delta(z)^s} \frac{G_s(y,z)}{\delta(z)^s} \langle z-\xi, \nu\rangle\,d\sigma = (N-2s) H_s(x,y) + \langle\nabla_x H_s(y,x), x-\xi\rangle + \langle\nabla_y H_s(x,y), y-\xi\rangle.$$ Taking $y = x$, this recovers a representation for the Robin function $\mathcal{R}_s(x) = H_s(x,x)$ in terms of a boundary integral, generalizing the classical result by Brezis and Peletier for $s=1$, and closely related to fractional capacity and concentration phenomena.

7. Extensions, Mixed Operators, and Future Directions

Recent work investigates Pohozaev identities for systems and operators combining local and nonlocal effects, e.g., combining anisotropic $p$-Laplacian and fractional $p$-Laplacian (Anthal et al., 10 Jun 2025). For a mixed problem

$$-\alpha H_p u + \beta (-\Delta_p)^s u = f(u) \quad \text{in} \;\; \mathbb{R}^n$$

the identity is

$$\frac{\alpha(n-p)}{p}\|H(\nabla u)\|_p^p + \frac{\beta(n-sp)}{p}[u]_{s,p}^p = n\int_{\mathbb{R}^n} F(u) dx,$$

with $F(u)=\int_0^u f(s)ds$. This mixed case identity, new even for $p=2$, synthesizes local and nonlocal scaling effects and allows nonexistence results and detailed qualitative analysis.

The evolution of Pohozaev-type identities also encompasses degenerate operators (Grushin, sub-Laplacian), conformally variational and nonlocal energies (half-harmonic maps), and flows (Ricci solitons, reaction-diffusion), often requiring fine regularity theory, domain variation arguments, and the use of adapted function spaces.

---

In summary, the Pohozaev type identity forms a fundamental analytic and geometric tool across diverse domains, rooted in the invariance and variational properties of the underlying physical or geometric system. Its universality arises from a deep interplay between symmetry, boundary conditions, and the structure of the associated energy or Lagrangian, allowing robust applications ranging from nonexistence and uniqueness to rigidity in geometric analysis, spectral theory, and nonlocal PDEs.