Struwe Type Global Compactness

• The result captures how bounded Palais–Smale sequences decompose into a weak limit and bubble profiles representing concentration phenomena.
• It employs profile decomposition techniques to quantify energy splitting and establish asymptotic orthogonality of scaled bubbles.
• Extensions include applications to nonlocal, fractional, and weighted Sobolev spaces, impacting existence and multiplicity proofs for critical equations.

The Struwe type global compactness result, originating from Struwe’s seminal work on Sobolev critical equations, describes the precise manner in which bounded Palais–Smale sequences may fail to exhibit compactness in variational problems with critical nonlinearities. Such loss of compactness is intrinsically associated with the formation of “bubble” profiles – rescaled solutions of the limiting problem, often corresponding to entire-space or boundary-concentrated solutions. Modern formulations adapt the original principle to wide-ranging operators and geometries, including fractional Laplacians, weighted Sobolev spaces, nonlocal functionals, hyperbolic and sub-Riemannian geometries, and flows for nonlinear PDEs. The principle is used to characterize defect measures, energy decomposition, and to prove existence and multiplicity results for critical equations, notably via profile decomposition technology for Palais–Smale sequences.

1. General Statement and Definitions

The framework begins with a Hilbert or Banach variational setting, typically involving a Sobolev-type space $X$ over a domain $\Omega$ (which may be $\mathbb{R}^N$, a bounded subset, a manifold, or a group such as the Heisenberg group), and an energy functional $I$ involving a critical nonlinearity. A Palais–Smale sequence $\{u_n\}\subset X$ at level $c$ for $I$ satisfies

$I(u_n) \to c, \quad I'(u_n) \to 0 \text{ in } X'.$

Such sequences may concentrate, and the decomposition provided by Struwe-type compactness identifies all possible defects.

The global compactness result asserts, up to subsequence, the following decomposition for each $u_n$:

$$u_n = u + \sum_{j=1}^J (\lambda_n^j)^{-a} W^j\left(\frac{x-x_n^j}{\lambda_n^j}\right) + r_n,$$

where $u$ is the weak limit in $X$, $W^j$ are nontrivial solutions to the limiting equation (on the whole space or half-space), $(x_n^j, \lambda_n^j)$ are (possibly diverging) centers and scales with asymptotic orthogonality, and $r_n \to 0$ strongly in $X$. The energy splits accordingly

$I(u_n) = I(u) + \sum_{j=1}^J I_\infty(W^j) + o(1).$

This structure is universal for critical problems—both local and nonlocal—under suitable compactness, monotonicity, and nonexistence assumptions for the limiting equations (Bhakta et al., 2023, Chakraborty et al., 22 Apr 2025, Brasco et al., 2016, Palatucci et al., 2023, Chernysh, 2024, Palatucci et al., 2014, Biswas, 2 Jan 2026).

2. Profile Decomposition and Energy Splitting

The core of Struwe-type compactness is profile decomposition—an analytic representation of loss of compactness as a sum of bubble profiles. These are typically Aubin–Talenti type functions in Euclidean cases,

$W(y) = [1 + |y|^2]^{-(N-2)/2},$

or Jerison–Lee extremals in the Heisenberg group,

$$\omega(\xi) = C(1 + |\xi|_{\mathbb{H}}^2)^{-(Q-2)/2}.$$

In fractional or weighted settings, bubbles are solutions to the corresponding limit equations involving fractional Laplacians or weights.

Energy and norm splitting results formalize quantization:

$$\|u_n\|^2_X = \|u\|^2_X + \sum_j \|W^j\|^2_X + o(1), \quad I(u_n) = I(u) + \sum_j I_\infty(W^j) + o(1),$$

with cross-terms vanishing due to asymptotic orthogonality of scales and centers (Palatucci et al., 2014, Mercuri et al., 2011, Bhakta et al., 2023, Brasco et al., 2016).

For nonlocal or fractional operators, one must employ profile decomposition in fractional Sobolev spaces and control interactions via Brezis–Lieb type lemmas, Caccioppoli inequalities, and precise scaling invariance results (Palatucci et al., 2014, Biswas, 2 Jan 2026, Chakraborty et al., 22 Apr 2025).

3. Geometric and Analytic Structures

Concentration phenomena often reflect the underlying geometry:

• In hyperbolic spaces, bubbles may “escape to infinity” along isometries (Möbius translations) and are described as hyperbolic bubbles $U_\infty \circ T_{a_n^j}^{-1}$—with $T_a$ the translation carrying $0 \mapsto a$ (Bhakta et al., 2023).
• In sub-Riemannian settings (Heisenberg group), the profile decomposition uses group dilations and translations, and bubbles can occur in the whole group or half-spaces tangent to the boundary, with scaling according to the group’s homogeneous dimension $Q$ (Palatucci et al., 2023, Palatucci et al., 2023).
• Weighted Sobolev spaces involve rescaling adapted to the weights (CKN-type transforms) to ensure bubble profiles are compatible with the functional’s structure (Chernysh, 2024).
• Nonlocality and fractional exponents modify the translation-dilation group and require estimates on fractional seminorms (Palatucci et al., 2014, Brasco et al., 2016, Biswas, 2 Jan 2026).

Orthogonality conditions on scales and centers (e.g., $$|\log(\lambda_n^i/\lambda_n^j)| + |x_n^i - x_n^j|/\lambda_n^i \to \infty$$) ensure decoupling of bubbles, preventing interaction in the limit.

4. Methodological Outline and Proof Ingredients

The standard proof proceeds via:

• Boundedness and weak convergence extraction, showing initial PS sequences have a weak limit solving the full problem.
• Iterated subtraction and rescaling to extract bubble profiles: concentration functions (Levy-type) are used to locate scales of concentration.
• Use of precise analytic lemmas: Brezis–Lieb splitting, Leoni interpolation, Caccioppoli inequalities, Vitali convergences, null-form estimates in wave-map contexts.
• Structure of non-interaction and energy quantization: minimal energy thresholds for each nontrivial bubble guarantee finite termination (Mercuri et al., 2011, Chiodaroli et al., 2016, Biswas, 2 Jan 2026).

In flows (parabolic or map flows), compactness away from the concentration set is obtained using monotonicity formulas and $\varepsilon$-regularity theorems, identifying the singular set with codimension at least $p$ (for $p$-harmonic maps), with strong convergence outside (Hupp et al., 2023).

5. Extensions and Applications

Struwe-type compactness underpins existence/multiplicity proofs for critical equations, blow-up analysis, and is essential for variational constructions where the (PS)-condition fails. Extensions include:

• Fractional and nonlocal operators, via fractional Gagliardo norm and nonlocal profile decomposition (Brasco et al., 2016, Palatucci et al., 2014, Biswas, 2 Jan 2026).
• Weighted Sobolev inequalities, with CKN-type bubbles and additional transform rules (Chernysh, 2024).
• Quasi-linear systems, exterior domains, and inclusion of lower-order terms with appropriate growth and coercivity (Mercuri et al., 2011).
• Sub-Riemannian spaces, CR–Yamabe problems, and concentration on Carnot groups, employing group-specific rescaling (Palatucci et al., 2023, Palatucci et al., 2023).
• Wave maps and geometric flows, where the decomposition occurs in the energy class for solutions of critical nonlinear wave equations (Chiodaroli et al., 2016).
• Parabolic flows: strong convergence away from the concentration set, with bounds on the Minkowski dimension of singularities (Hupp et al., 2023).

6. Geometric, Topological, and Functional Consequences

The identification of “bubbles” as all possible sources of defect in compactness facilitates:

• Precise description of blow-up and quantitative concentration phenomena.
• Existence and multiplicity results via topological methods (e.g., the Coron–Bahri–Coron argument, using nontrivial loops in solution space and domain topology) (Chakraborty et al., 22 Apr 2025).
• Classification of limiting problems, uniqueness or non-existence results for profile equations (Pohozaev-type identities), and criteria for the presence of boundary or half-space bubbles (Palatucci et al., 2023, Palatucci et al., 2014).
• Quantization of blow-up mass, notably in dimension 2 for exponential nonlinearities, with fixed quantization at boundary points (Bahoura, 2018).

7. Summary Table: Key Ingredients Across Geometries

Setting	Bubble Profiles	Operator(s)
Euclidean	Aubin–Talenti functions	$-\Delta$, $(-\Delta)^s$, $-\Delta_p$
Hyperbolic space	$U_\infty \circ T_a^{-1}$	Laplace–Beltrami
Heisenberg group	Jerison–Lee extremals	sub-Laplacian $\Delta_H$
Weighted spaces (CKN)	Weighted critical bubbles	Weighted $p$-Laplacian
Fractional/Nonlocal	Fractional bubbles	$(-\Delta)^s$, $(-\Delta_p)^s$
Wave maps, flows	Energy profiles in scaling	Geometric PDE, parabolic flows

Each context employs the general principle that any lack of compactness in critical problems is precisely characterized by the emergence of nontrivial bubble profiles, rescaled and located at distinct scales and centers, whose parameters become asymptotically orthogonal and whose energies sum with a vanishing remainder, as formalized in the Struwe type global compactness result (Bhakta et al., 2023, Chakraborty et al., 22 Apr 2025, Brasco et al., 2016, Palatucci et al., 2023, Chernysh, 2024, Palatucci et al., 2014, Biswas, 2 Jan 2026, Chiodaroli et al., 2016, Mercuri et al., 2011, Bahoura, 2018, Palatucci et al., 2023, Hupp et al., 2023).