Pohozaev Manifold in Nonlinear PDEs

• Pohozaev manifolds are codimension-one C¹ submanifolds defined as the zero level set of a Pohozaev functional, encapsulating scaling invariance.
• They provide a natural constraint in variational frameworks, ensuring existence, uniqueness, and regularity of ground states for nonlinear PDEs.
• They extend to fractional, higher-order, and coupled systems, enabling projection techniques and robust numerical methods for ground-state search.

A Pohozaev manifold is a codimension-one $C^1$ submanifold of a Hilbert or Banach function space, defined as the zero level set of the Pohozaev functional associated to a nonlinear partial differential equation (PDE) or variational problem. The manifold encapsulates the nontrivial constraints arising from the Pohozaev identity—a scalar first integral obtained through the multiplication of the PDE by a conformal or dilation vector field and integration by parts. This constraint encodes scaling invariance or its generalizations, and enforces restrictions on admissible solutions, often distinguishing existence regimes, regularity, and uniqueness of ground states. The Pohozaev manifold forms the geometric backbone of constrained variational methods, providing a natural constraint for finding critical points of energy-type functionals. It plays a decisive role in the analysis of scalar equations, coupled systems, fractional and higher-order Laplacians, Schrödinger equations, and nonlinear elliptic systems.

1. Construction of the Pohozaev Manifold

The construction of a Pohozaev manifold begins with the derivation of the Pohozaev identity for a PDE of the archetypal form

$$-\Delta u + \lambda u = f(u) \quad \text{in } \mathbb{R}^N, \quad u \in H^1(\mathbb{R}^N), \quad \lambda > 0, \ N \geq 3.$$

Multiplying the equation by $x\cdot\nabla u$ and integrating yields (see (Maia et al., 2019)): $$(N-2) \int_{\mathbb{R}^N} |\nabla u|^2 \, dx = 2N \int_{\mathbb{R}^N} G(u) \, dx,$$ where $G(u) = -\frac{\lambda}{2} u^2 + F(u)$, $F(u) = \int_0^u f(t)dt$.

One then defines the Pohozaev functional $P:H^1(\mathbb{R}^N)\rightarrow\mathbb{R}$: $$P(u) = \int_{\mathbb{R}^N} |\nabla u|^2 dx - \frac{2N}{N-2}\int_{\mathbb{R}^N}G(u) dx,$$ and the associated Pohozaev manifold as

$$\mathcal{M} = \{\, u \in H^1(\mathbb{R}^N) \setminus \{0\} \;|\; P(u) = 0 \, \}.$$

Under suitable monotonicity and regularity conditions on $f$, $\mathcal{M}$ is a closed $C^1$ hypersurface, with $\|u\|_{H^1} \geq \sigma > 0$ for $u \in \mathcal{M}$ (see (Maia et al., 2019, Maia et al., 2020)).

2. Functional-Analytic and Geometric Properties

The Pohozaev manifold exhibits several robust geometric and analytic features:

• Codimension-One Structure: As the zero set of a $C^1$ functional whose derivative does not vanish tangentially (regular value property), $\mathcal{M}$ is a smooth hypersurface in the ambient Hilbert space.
• Uniform Coercivity: There exists $\sigma>0$ such that all elements $u\in\mathcal{M}$ satisfy $\|u\|_{H^1}\ge\sigma$, precluding collapse to zero and ensuring robustness for variational methods (Maia et al., 2019, Maia et al., 2020).
• Natural Constraint: Any constrained critical point of the energy functional on $\mathcal{M}$ is a genuine free critical point (Lagrange multiplier vanishes), making $\mathcal{M}$ a natural constraint (Liu et al., 1 Dec 2024, Albuquerque et al., 14 Nov 2025).
• Projection Mechanisms: For trial functions, projection onto the manifold via scaling or dilation yields unique representatives within $\mathcal{M}$ along rays, facilitating both analytical and numerical ground-state search (Maia et al., 2019, Maia et al., 2020).

3. Variational Role and Ground-States

The principal variational framework seeks minimizers or saddle points of an energy functional under the Pohozaev constraint. For instance, in

$$I(u) = \frac{1}{2}\int_{\mathbb{R}^N}(|\nabla u|^2 + \lambda u^2) dx - \int_{\mathbb{R}^N} F(u) dx,$$

critical points on $\mathcal{M}$ correspond to weak solutions of the original PDE (Maia et al., 2019). The mountain-pass level, the ground-state minimization, and their equivalence on $\mathcal{M}$, are established via minimax theory: $$c = \inf_{\gamma \in \Gamma} \max_{t \in [0,1]} I(\gamma(t)), \quad m = \inf \{I(u): u \neq 0, I'(u) = 0\}, \quad m_P = \inf_{u \in \mathcal{M}} I(u),$$ with $m = c = m_P$ under suitable conditions (Maia et al., 2019).

Extending the concept, the Nehari–Pohozaev manifold for coupled systems or fractional equations investigates constraints arising from both the usual Nehari functional and the Pohozaev identity (Albuquerque et al., 14 Nov 2025, Bueno et al., 2018).

4. Extensions: Fractional and Higher-Order Operators

For nonlocal operators such as the (higher-order) fractional Laplacian $(-\Delta)^s$, $s > 0$, the Pohozaev manifold analogue is defined through an identity including a boundary term involving nonlocal traces: $$P(u) := (2s - n)\! \int_{\Omega} u f(u) \, dx + 2n\! \int_{\Omega} F(u) \, dx - \Gamma(1+s)^2 \int_{\partial\Omega} \left(\frac{u}{d^s}\right)^2 (x \cdot \nu) \, d\sigma,$$ with associated manifold

$$\mathcal{M} = \{\, u \in H_0^s(\Omega) \setminus \{0\} : P(u) = 0 \, \}.$$

Such definitions carry over to pseudo-relativistic operators and to cases with critical or supercritical exponents, often determining precise existence and nonexistence regimes (Ros-Oton et al., 2012, Ros-Oton et al., 2014, Ros-Oton et al., 2012, Bueno et al., 2018).

5. Numerical and Algorithmic Implementation

The Mini-Max Algorithm via the Pohozaev Manifold (MMAP) typifies the application of the Pohozaev projection in computational settings (Maia et al., 2019):

1. Initialization: Select $w_0$ so that $\int G(w_0) > 0$.
2. Pohozaev Projection: For $u$, find unique $t^*$ so that $u_1(x) = u(x / t^*) \in \mathcal{M}$, with $t^*$ computed via explicit formula.
3. Descent Direction: Solve the linearized Euler–Lagrange equation for the normalized descent direction.
4. Line Search/Re-projection: Search along descent direction, re-projecting onto $\mathcal{M}$ and minimizing energy.
5. Update/Iteration: Iterate until convergence under the Palais–Smale condition.

This approach ensures convergence to ground states in situations where traditional Nehari manifold methods may fail, especially for asymptotically linear or superlinear nonlinearities.

6. Applications to PDE Systems, Schrödinger Equations, and Constraints

Pohozaev manifolds generalize naturally:

• Hartree–Fock Systems: Employ Nehari–Pohozaev manifolds incorporating interaction terms and multiple variables (Albuquerque et al., 14 Nov 2025).
• Nonlinear Schrödinger with Mass Constraint: $L^2$-Pohozaev manifolds ensure existence and multiplicity of solutions under critical thresholds for interaction parameters (Liu et al., 1 Dec 2024).
• Equations with Vanishing Potentials: Projection onto the Pohozaev manifold overcomes zero-mass and lack of spectral gap for solutions in unbounded domains (Maia et al., 2020).
• Geometric and Physical Problems: The Pohozaev–Schoen identity on Riemannian manifolds unifies scalar curvature prescription and energy-momentum conservation (Gover et al., 2010).

7. Impact on Existence, Nonexistence, and Uniqueness Results

The Pohozaev manifold forms the foundation for:

• Existence of Ground States: Constrained minimization on the manifold yields unique, radially symmetric ground states under monotonicity and compactness hypotheses (Maia et al., 2019, Liu et al., 1 Dec 2024).
• Nonexistence in Supercritical Regimes: If the nonlinear term surpasses a critical threshold, the Pohozaev identity cannot be satisfied, and $\mathcal{M}$ is empty, precluding nontrivial solutions (Ros-Oton et al., 2012, Bueno et al., 2018).
• Multiplicity and Bifurcation: Variational bifurcation techniques on submanifolds split by local minimization and mountain-pass critical points distinguish multiple positive solutions (Liu et al., 1 Dec 2024).
• Symmetry and Regularity: Pohozaev constraints, sometimes combined with moving-plane methods, enforce spherical symmetry and regularity properties in solutions (Maia et al., 2019, Bueno et al., 2018).

The Pohozaev manifold thus serves as a critical analytic and geometric construct in modern nonlinear PDE theory, variational analysis, and computational methods. Its multi-faceted impact spans and connects domains from geometric analysis and mathematical physics to high-dimensional numerical computation.