On Coron problems with Choquard term and mixed operator

Abstract: In this article, we study a Coron-type problem involving a critical Choquard nonlinearity driven by a mixed operator combining the Laplacian and fractional Laplacian. In annular-type domains, we prove the existence of nontrivial positive solutions when the inner hole is sufficiently small. Using variational methods and concentration compactness arguments, we establish a global compactness result for Palais- Smale sequences and obtain high-energy solutions using topological methods. We also derive regularity results for weak solutions.

• The paper demonstrates existence of positive high-energy solutions to critical Choquard equations with a mixed operator combining Laplacian and fractional Laplacian in annular domains.
• It develops a novel global compactness theorem and profile decomposition technique, overcoming challenges from dual nonlocality and critical HLS nonlinearity.
• The study employs a topological argument inspired by Coron’s method to ensure that the variational min-max level is achieved below the next compactness threshold.

Coron Problems for Critical Choquard Equations with Mixed Local-Nonlocal Operators

Introduction and Problem Setting

This work investigates the existence, compactness, and regularity of positive high-energy solutions to a critical elliptic equation of Choquard type with a mixed local-nonlocal operator, namely the sum of the Laplacian and the fractional Laplacian. The PDE takes the form

$$\mathcal{L}u(x) = \left(\int_{\Omega} \frac{u(y)^{2^*_\mu}}{|x-y|^\mu}dy\right) u(x)^{2^*_\mu-1}$$

where $\mathcal{L} = -\Delta + (-\Delta)^s$ with $s \in (0,1)$, and the nonlinearity involves the upper critical exponent for the Hardy-Littlewood-Sobolev (HLS) convolution, $2^*_\mu = (2n-\mu)/(n-2)$. The domain $\Omega$ is assumed to be an annular-type region in $\mathbb{R}^n$.

The prototype is a mixed local-nonlocal Brezis-Nirenberg-type problem at Sobolev criticality, but where the nonlinearity is replaced by the nonlocal critical Choquard term. The primary analytic difficulties stem from the double lack of compactness induced by the mixed operator and the critical HLS nonlinearity.

Mathematical Framework and Compactness Theory

The analysis is performed in the Hilbert space $\mathcal{X}_0^{1,2}(\Omega)$, the natural energy space associated with the mixed operator, equipped with the norm

$$\eta(u) = \left( \|\nabla u\|^2_{L^2} + [u]_s^2 \right)^{1/2}$$

where $[u]_s$ denotes the Gagliardo seminorm. The main variational functional is

$$I(u) = \frac{1}{2} \left( \int |\nabla u|^2 + [u]_s^2 \right) - \frac{1}{2\cdot 2^*_\mu} \iint_{\Omega\times\Omega} \frac{|u(x)|^{2^*_\mu}|u(y)|^{2^*_\mu}}{|x-y|^\mu}dxdy$$

which is $\mathcal{L} = -\Delta + (-\Delta)^s$0 on $\mathcal{L} = -\Delta + (-\Delta)^s$1 due to the HLS inequality and Sobolev embeddings.

A significant technical achievement is a global compactness theorem of Struwe-type for Palais-Smale sequences of $\mathcal{L} = -\Delta + (-\Delta)^s$2, featuring a decomposition into solution profiles at infinity. Crucially, the lack of an invariant scaling group for the mixed operator complicates the analysis compared to pure Laplacian or pure fractional problems. To overcome this, the authors introduce a Morrey-type concentration analysis coupled with precise energy decoupling using Brezis-Lieb-type results on the nonlocal Choquard term.

Concentration Profile Decomposition

Suppose $\mathcal{L} = -\Delta + (-\Delta)^s$3 is a Palais–Smale sequence for $\mathcal{L} = -\Delta + (-\Delta)^s$4. Then, up to subsequence, there exists a solution $\mathcal{L} = -\Delta + (-\Delta)^s$5 of the problem on $\mathcal{L} = -\Delta + (-\Delta)^s$6 and at most $\mathcal{L} = -\Delta + (-\Delta)^s$7 nontrivial solutions $\mathcal{L} = -\Delta + (-\Delta)^s$8 of the limiting problem on $\mathcal{L} = -\Delta + (-\Delta)^s$9 such that

$s \in (0,1)$0

with separation of centers and scales, and the energies add up: $s \in (0,1)$1 where $s \in (0,1)$2 is the corresponding energy functional for the limiting problem.

This sharp decomposition is critical for addressing the lack of compactness and analyzing the existence of critical points above the ground state level.

Existence of High-Energy Solutions in Annular Domains

The main existence result is the realization of the so-called "Coron phenomenon" for the critical mixed Choquard problem, i.e., the recovery of compactness and existence of positive high-energy solutions in domains with a small inner hole (annular or non-contractible domains). The authors adapt the classical two-bubble topology deformation strategy (Coron, Bahri-Coron) to the nonlocal critical context.

By constructing a two-parameter family of Aubin-Talenti bubbles localized via cut-off functions and analyzing their energies with respect to the Sobolev-type quotient, the authors show that for $s \in (0,1)$3 large enough and the annulus sufficiently thick, the max energy of these bubbles remains strictly below the next compactness threshold. The crucial estimate is: $s \in (0,1)$4 where $s \in (0,1)$5 is the best constant in the HLS-Sobolev inequality, and $s \in (0,1)$6 is independent of $s \in (0,1)$7 and $s \in (0,1)$8. This ensures the variational min-max level is achieved below the second critical value.

A careful topological argument and the precise construction of a retraction from the configuration space of bubbles to the sphere in the domain, forbidden by the topology (hole), then guarantee the existence of a positive high-energy solution.

Regularity and Maximum Principles

The paper establishes optimal regularity of weak solutions using De Giorgi-type iterations and bootstrapping regularity results for mixed operators from the literature, showing any weak solution is in $s \in (0,1)$9 and $2^*_\mu = (2n-\mu)/(n-2)$0 for suitable choices of the parameters.

A strong maximum principle tailored to mixed operators is proved, guaranteeing the positivity of nontrivial solutions and supporting the validity of the topological arguments in the existence proof.

Energy Estimates and Non-Attainment of Extremals

A notable feature of the mixed Choquard problem is that the best Sobolev-HLS constant $2^*_\mu = (2n-\mu)/(n-2)$1 is never attained in bounded or annular domains, but only on the whole space, by explicit Aubin-Talenti profiles. This non-attainment is central to the appearance of bubbling/concentration phenomena, and thereby makes profile decomposition and high-energy critical points the only possible solutions in the critical regime. The energy of any sign-changing or positive solution is bounded below by an explicit function of $2^*_\mu = (2n-\mu)/(n-2)$2, which sets the ground state and first energy level.

Implications and Perspectives

The results provide several key theoretical advances:

• Extension of the global compactness (profile decomposition) framework to mixed (local + nonlocal) critical problems with HLS-type convolution nonlinearities.
• Resolution of a Coron-type problem for the mixed critical Choquard equation: topological nontriviality (“hole”) can restore compactness, enabling the existence of positive high-energy solutions in contrast with star-shaped (contractible) domains.
• A variational-topological contradiction argument at the critical level, combining analytic concentration-compactness with geometric information.

From an applied viewpoint, such mixed models naturally arise in models featuring simultaneous local and anomalous (Lévy) diffusion, as encountered in mathematical physics, biology (animal mobility), and population dynamics.

Future Directions

The analysis opens several lines for further research:

• Extension to systems (vector-valued) of mixed critical Choquard equations.
• Multiplicity and qualitative properties (symmetry breaking, nodal solutions).
• Mixed problems with other types of nonlocal nonlinearities or different boundary conditions.
• Fine dynamics and stability of the bubbling/high energy solutions constructed.

Conclusion

The paper rigorously establishes the existence of positive high-energy solutions for critical Choquard equations involving the sum of the Laplacian and a fractional Laplacian under annular geometric conditions, by a sharp variational and concentration-compactness analysis. It significantly advances the understanding of the interplay between domain topology, mixed differential operators, and nonlocal critical nonlinearities. The approach and results set a foundation for further exploration of mixed-operator problems with nonlocal critical nonlinearities and their applications.