Critical Hardy-Sobolev Exponent Problems

• Nonlinear critical Hardy-Sobolev exponent problems are elliptic and nonlocal equations characterized by critical growth, incorporating Hardy potentials and weighted nonlinearities.
• Variational methods—using tools like Nehari manifolds and mountain-pass theorems—establish the existence, bifurcation, and multiplicity of ground and bound state solutions.
• Analytical techniques such as concentration-compactness, refined Hardy-Sobolev inequalities, and profile decomposition address loss of compactness near singularities.

Nonlinear Critical Hardy-Sobolev Exponent Problems comprise a class of elliptic and nonlocal equations and systems that exhibit critical growth combining the Hardy potential with Sobolev admissible and weighted nonlinearities. These problems are characterized by intricate phenomena regarding variational compactness, concentration of Palais-Smale sequences, and bifurcation of ground and bound state solutions, typically in functional frameworks involving fractional Laplacians, coupling potentials, and critical Hardy-Sobolev type exponents. Their study has evolved to include scalar PDEs, elliptic systems, nonlocal boundary-value problems, and fractional and quasilinear variants.

1. Mathematical Framework and Critical Exponents

Critical Hardy-Sobolev problems are set in fractional Sobolev spaces $D^{s,2}(\mathbb{R}^N)$ for $0 < s < 1$, or classical $H^1_0(\Omega)$, with energy involving the Gagliardo semi-norm and Hardy-type weights. The fractional Laplacian $(-\Delta)^s$ or its nonlocal analogs act as the principal operator.

For a prototypical fractional system, model equations admit the form: $$\begin{cases} (-\Delta)^s u - \frac{\mu_1 u}{|x|^{2s}} = \frac{|u|^{2^*_{s,t}-2}u}{|x|^t} + \nu h(x) \alpha u^{\alpha-1} v^\beta, \ (-\Delta)^s v - \frac{\mu_2 v}{|x|^{2s}} = \frac{|v|^{2^*_{s,t}-2}v}{|x|^t} + \nu h(x) \beta u^\alpha v^{\beta-1}, \end{cases}$$ with $2^*_{s,t} = \frac{2(N-t)}{N-2s}$ the fractional Hardy-Sobolev critical exponent, and $h \in L^{2^*_{s,t}/(\alpha+\beta)}$ a weight (Ortega, 2023).

Weighted $L^p$ spaces $L^p_t(\mathbb{R}^N)$ are used to encode singularity and critical growth: $$L^p_t(\mathbb{R}^N) = \left\{u : \int_{\mathbb{R}^N} \frac{|u(x)|^p}{|x|^t} \, dx < \infty \right\}.$$

Scaling invariance at criticality is preserved by $u(x) \mapsto \lambda^{(N-2s)/2} u(\lambda x)$, precisely for the exponent relation above.

2. Variational Structure and Nehari Geometry

Variational methods are crucial. The canonical energy functional, for scalar equations or systems, is: $$J_\nu(u,v) = \frac{1}{2} \left( \|u\|_{1,s}^2 + \|v\|_{1,s}^2 \right) - \frac{1}{2^*_{s,t}} \int \frac{|u|^{2^*_{s,t}} + |v|^{2^*_{s,t}}}{|x|^t} - \nu \int h(x) u^\alpha v^\beta.$$ Critical points are restricted to the Nehari manifold $$\mathcal N_\nu = \{(u,v) \ne (0,0): \langle J_\nu'(u,v), (u,v) \rangle = 0\}$$, which is a $C^1$ manifold of codimension one (Ortega, 2023).

On $\mathcal N_\nu$, $J_\nu$ is bounded below and coercive; ground states correspond to global minimizers on $\mathcal N_\nu$, while mountain-pass geometry yields bound-state solutions in regimes where bifurcation occurs.

For scalar equations, analogous constructions are used, with the functional form adapted to incorporate convolution terms and Hardy-Sobolev weights (Gu et al., 19 Sep 2025, Zhong et al., 2015, Yang, 2017).

3. Loss of Compactness and Concentration-Compactness Principles

At the critical Hardy-Sobolev exponent, conventional Sobolev embeddings lose compactness. As a result, Palais-Smale sequences for variational problems may concentrate by forming "bubbles" either near the singularity at $x=0$ or at infinity.

A fractional concentration-compactness lemma (cf. Lions) guarantees that any $(PS)_c$ sequence decomposes into a strong limit plus at most one bubble at $0$ or at infinity. Compactness of $(PS)_c$ sequences is recovered whenever the energy level $c$ stays below the threshold $$\mathfrak{c}(\mu) = \frac{2s-t}{2(N-t)} [S(\mu,t,s)]^{(N-t)/(2s-t)}$$, where $S(\mu,t,s)$ is the best constant in the Hardy-Sobolev inequality (Ortega, 2023). Similar results hold for local, nonlocal, and quasilinear operators and for doubly or multiply critical settings (Biswas, 2 Jan 2026, Su et al., 2018, Perera et al., 2016).

Refined embedding theorems, such as endpoint- or interpolation-type Hardy-Sobolev inequalities, are pivotal to control loss of compactness and ensure the non-vanishing of critical norm terms (Su et al., 2018, Su et al., 2018).

4. Existence, Multiplicity, and Bifurcation of Solutions

Existence results bifurcate according to the strength of coupling parameters and the arrangement of exponents:

• Large coupling $\nu$: ground states are genuinely coupled, with energy below all semi-trivial branches.
• Small coupling $\nu$: the ground state coincides with the minimizing semi-trivial solution (either $(z_1,0)$ or $(0,z_2)$, extremal for $S(\mu_i,t,s)$).
• Intermediate or interlacing energies: mountain-pass paths constructed in the Nehari geometry yield bound states distinct from semi-trivial solutions.

Regimes of existence (for scalar and system cases) depend on the sum $\alpha+\beta$ relative to $2^*_{s,t}$, and technical assumptions on the weight $h$ (vanishing at $0$, infinity, or appropriate integrability) when $\alpha+\beta=2^*_{s,t}$ (Ortega, 2023, López-Soriano et al., 2022).

Multiplicity is obtained by linking arguments, cohomological index theory, and spectral separation, particularly in p-Laplacian and biharmonic settings (Yang, 2017, Drissi et al., 2023, Perera et al., 2016). Bifurcation analysis discovers parameter thresholds $\nu_0$ at which semi-trivial branches lose minimization character and nontrivial coupled solutions bifurcate (Ortega, 2023, López-Soriano et al., 2022, Zhong et al., 2015).

Systems and equations admitting multiple critical nonlinearities—Hardy, Sobolev, and Hardy-Littlewood-Sobolev (HLS)—require sharp test-function constructions and energy level estimates to establish existence and distinguish solution branches (Su et al., 2018, Gu et al., 19 Sep 2025, Li et al., 2011).

5. Technical Tools and Key Estimates

Critical Hardy-Sobolev problems rely on:

• Sharp Hardy-Sobolev and HLS inequalities: These provide the foundational embedding constants $S(\mu,t,s)$, $S_{H,L}$, and sharp profiles required for energy comparison and compactness control (Su et al., 2018, Su et al., 2018, Gao et al., 2016).
• Profile decomposition: Bubble profiles are characterized as extremals of the relevant best-constant inequality (radially decreasing solutions, Aubin-Talenti type, fractional analogs) (Ortega, 2023, Yang, 2017, Biswas, 2 Jan 2026).
• Energy splitting and Brezis-Lieb lemma: These enable precise energy decompositions along PS sequences and validate the limit passage between sequences and bubble components (Biswas, 2 Jan 2026, Gao et al., 2016, Su et al., 2018).
• Nehari manifold methods: Minimax and critical-point theorems are implemented on tailored constraint manifolds to produce ground and bound states (Zhong et al., 2015, López-Soriano et al., 2022, Ortega, 2023).
• Concentration control via refined inequalities: Interpolated Hardy-Sobolev inequalities and Morrey-type bounds exclude dichotomy and vanishing phenomena in Palais-Smale analysis (Su et al., 2018, Su et al., 2018).

6. Generalizations, Open Problems, and Research Directions

Recent extensions address:

• Fractional and p-Laplacian operators: Including doubly and multiply critical settings, further generalized to systems and nonlocal convolution nonlinearities (Ortega, 2023, Biswas, 2 Jan 2026, Yang, 2017).
• Critical elliptic systems: Nonlinear coupling through critical Hardy-Sobolev and HLS exponents, with ground and bound state solutions parameterized by coupling constants and exponent arrangements (López-Soriano et al., 2022, Giacomoni et al., 2017, Hong et al., 2018).
• Interaction of local and nonlocal critical nonlinearities: Problems featuring combined Hardy-Sobolev and HLS critical growth reveal delicate compactness and energy tradeoffs (Gu et al., 19 Sep 2025, Gao et al., 2016, Su et al., 2018).
• Open problems: Explicit formulas for minimizers in fractional Hardy-Sobolev inequalities, existence and structure of sign-changing/nodal solutions, generalizations to anisotropic operators, less restrictive parameter ranges, and parabolic flow stability remain active research directions (Yang, 2017, Ortega, 2023).

The unification achieved by the variational approach, concentration-compactness, and energy level analysis across scalar equations, coupled systems, local and nonlocal operators, and fractional settings demonstrates a robust methodology and exposes deep links between critical growth, symmetry, and singularity at the heart of nonlinear Hardy-Sobolev theory.