On the existence results for $m$-Harmonic equation with critical Choquard Nonlinearity

Abstract: This article established the existence results for the $m$-harmonic equation involving critical Choquard nonlinearity and subcritical perturbation. We first explore the minimizers of the $m$-harmonic operator with the critical Choquard equation. Then, using these minimizers, we establish delicate estimates to show the energy below the threshold level, which helps to recover the compactness. Further, we prove the existence of a nontrivial solution for our problem with different kinds of local and nonlocal subcritical perturbations. To the best of our knowledge, this is the first article dealing with the polyharmonic equation and critical Choquard type nonlinearity. The results obtained are even new for $m\geq 2$.

• The paper establishes existence of nontrivial solutions for the m-harmonic equation with critical Choquard nonlinearity through rigorous variational methods.
• It computes sharp compactness thresholds and constructs explicit minimizers, generalizing Hardy–Littlewood–Sobolev inequalities for higher-order operators.
• The analysis combines mountain-pass and spectral techniques to address nonlocal perturbations, delineating precise parameter regimes for solution existence.

Existence Results for the Polyharmonic Equation with Critical Choquard Nonlinearity

Introduction and Motivation

This paper addresses the existence of nontrivial solutions to a class of elliptic PDEs involving the polyharmonic operator $(-\Delta)^m$, with a focus on equations exhibiting critical Choquard (nonlocal Hartree-type) nonlinearity on bounded domains $\Omega \subset \mathbb{R}^N$. Such problems extend classical results on both the polyharmonic (higher-order) equations and the nonlocal Choquard equations, synthesizing two technically demanding frameworks: the analysis of higher-order variational operators and the compactness issues intrinsic to critical nonlocal (convolution-type) nonlinearities. The work aims to establish minimization structures, sharp thresholds for compactness, and mountain-pass solutions in various parameter regimes for polyharmonic Choquard problems with superlinear local or nonlocal subcritical perturbations.

Mathematical Formulation

The principal equation studied is

$$(-\Delta)^m u = \left(\int_{\Omega} \frac{|u(y)|^{2_{\alpha,m}^*}}{|x-y|^{\alpha}} dy \right) |u(x)|^{2_{\alpha,m}^*-2}u(x) + \lambda f(u) \quad \text{in } \Omega,$$

with Dirichlet boundary conditions for derivatives up to order $m-1$. Here, $N \geq 2m+1$, $0<\alpha<N$, and the critical exponent in the sense of the Hardy--Littlewood--Sobolev inequality is $2_{\alpha,m}^* = \frac{2N-\alpha}{N-2m}$. The nonlinearity $f(u)$ is considered either as a local power-type superlinear term ($|u|^{q-2}u$) or a nonlocal convolution-type term.

The polyharmonic operator $\Delta^m u$ generalizes the Laplacian, enabling the investigation of higher-order effects pertinent in physics and geometry.

Variational and Analytical Framework

The variational formulation is central: solutions correspond to critical points of a functional on the Hilbert space $\Omega \subset \mathbb{R}^N$0, defined by the Dirichlet boundary conditions. The nonlocal term is handled with the Hardy–Littlewood–Sobolev (HLS) framework, leading to a critical exponent where the loss of compactness issues are most delicate, mirroring the classical Sobolev embedding at the critical exponent.

Crucially, the authors identify the best constant $\Omega \subset \mathbb{R}^N$1 for the polyharmonic HLS-type embedding and establish the structure of its minimizers, generalizing known extremals for the (bi)Laplacian case. The energy functional is then shown to admit a mountain-pass geometry, and careful energy and compactness threshold estimates are constructed to cope with the lack of compactness.

Summary of Main Results

Three main existence theorems are established for problem variants with local and nonlocal subcritical perturbations:

1. Superlinear Local Perturbations: For $\Omega \subset \mathbb{R}^N$2, $\Omega \subset \mathbb{R}^N$3, the authors show the existence of nontrivial solutions depending on the interplay between dimension $\Omega \subset \mathbb{R}^N$4, order $\Omega \subset \mathbb{R}^N$5, exponent $\Omega \subset \mathbb{R}^N$6, and the parameter $\Omega \subset \mathbb{R}^N$7. When $\Omega \subset \mathbb{R}^N$8 is large (w.r.t. $\Omega \subset \mathbb{R}^N$9), existence holds for all $$(-\Delta)^m u = \left(\int_{\Omega} \frac{|u(y)|^{2_{\alpha,m}^*}}{|x-y|^{\alpha}} dy \right) |u(x)|^{2_{\alpha,m}^*-2}u(x) + \lambda f(u) \quad \text{in } \Omega,$$0; for smaller $$(-\Delta)^m u = \left(\int_{\Omega} \frac{|u(y)|^{2_{\alpha,m}^*}}{|x-y|^{\alpha}} dy \right) |u(x)|^{2_{\alpha,m}^*-2}u(x) + \lambda f(u) \quad \text{in } \Omega,$$1, $$(-\Delta)^m u = \left(\int_{\Omega} \frac{|u(y)|^{2_{\alpha,m}^*}}{|x-y|^{\alpha}} dy \right) |u(x)|^{2_{\alpha,m}^*-2}u(x) + \lambda f(u) \quad \text{in } \Omega,$$2 must be sufficiently large.
2. Superlinear Nonlocal Perturbations: For subcritical convolution-type $$(-\Delta)^m u = \left(\int_{\Omega} \frac{|u(y)|^{2_{\alpha,m}^*}}{|x-y|^{\alpha}} dy \right) |u(x)|^{2_{\alpha,m}^*-2}u(x) + \lambda f(u) \quad \text{in } \Omega,$$3, analogous results quantify thresholds in parameters for the existence of solutions and detail compactness restoration below an explicit energy threshold.
3. Thresholds and Compactness: In each variational setting, the mountain-pass critical value is shown to fall below the critical energy compactness threshold, ensuring that Palais–Smale sequences are precompact with respect to weak convergence in $$(-\Delta)^m u = \left(\int_{\Omega} \frac{|u(y)|^{2_{\alpha,m}^*}}{|x-y|^{\alpha}} dy \right) |u(x)|^{2_{\alpha,m}^*-2}u(x) + \lambda f(u) \quad \text{in } \Omega,$$4.

A detailed spectral analysis relates the role of $$(-\Delta)^m u = \left(\int_{\Omega} \frac{|u(y)|^{2_{\alpha,m}^*}}{|x-y|^{\alpha}} dy \right) |u(x)|^{2_{\alpha,m}^*-2}u(x) + \lambda f(u) \quad \text{in } \Omega,$$5 to the spectrum of $$(-\Delta)^m u = \left(\int_{\Omega} \frac{|u(y)|^{2_{\alpha,m}^*}}{|x-y|^{\alpha}} dy \right) |u(x)|^{2_{\alpha,m}^*-2}u(x) + \lambda f(u) \quad \text{in } \Omega,$$6 under Dirichlet boundary conditions, producing exact conditions for solutions when $$(-\Delta)^m u = \left(\int_{\Omega} \frac{|u(y)|^{2_{\alpha,m}^*}}{|x-y|^{\alpha}} dy \right) |u(x)|^{2_{\alpha,m}^*-2}u(x) + \lambda f(u) \quad \text{in } \Omega,$$7 (linear perturbation), and demonstrating that the existence regime shifts in low dimensions or for large $$(-\Delta)^m u = \left(\int_{\Omega} \frac{|u(y)|^{2_{\alpha,m}^*}}{|x-y|^{\alpha}} dy \right) |u(x)|^{2_{\alpha,m}^*-2}u(x) + \lambda f(u) \quad \text{in } \Omega,$$8.

Notably, the paper introduces, quantifies, and constructs explicit minimizers for the polyharmonic HLS inequality in $$(-\Delta)^m u = \left(\int_{\Omega} \frac{|u(y)|^{2_{\alpha,m}^*}}{|x-y|^{\alpha}} dy \right) |u(x)|^{2_{\alpha,m}^*-2}u(x) + \lambda f(u) \quad \text{in } \Omega,$$9. The minimal energy is directly related to the best constant for the embedding, and minimizers are explicitly exhibited as optimizers for the convolution structure.

Technical Innovations

• First Polyharmonic-Critical Choquard Results:

The analysis establishes existence for the $m-1$0-harmonic equation with critical nonlocal Choquard nonlinearity for the first time, including the explicit identification of minimizers.

• Threshold Computations:

The authors compute precise expansions of the minimizers' energies, and their delicate asymptotic behaviors, allowing rigorous estimates that confirm the compactness threshold falls strictly above the variational level for carefully constructed test functions.

• Nonlocal Brezis–Lieb Lemma:

A nonlocal analog of the Brezis–Lieb lemma is leveraged to facilitate splitting arguments in the presence of critical nonlocal nonlinearities, generalizing previous arguments to polyharmonic contexts.

• Spectral and Linking Arguments:

The existence proofs adapt both mountain pass and spectral (linking) methods, particularly to address indefinite functionals when the perturbation parameter $m-1$1 passes through eigenvalues.

Numerical and Qualitative Claims

The paper claims that for each regime (local or nonlocal perturbation), the constructed mountain-pass level $m-1$2 satisfies

$m-1$3

restoring the compactness required for the existence of weak solutions.

Particularly robust is the confirmation that the existence thresholds and spectral intervals for $m-1$4 are sharp with respect to the known structure for $m-1$5 and $m-1$6, and genuinely new for $m-1$7.

Implications and Future Directions

The results substantially broaden the theory of nonlinear elliptic equations involving critical nonlocal effects and higher-order operators. The analytical approach and variational construction lay the groundwork for further work on multiplicity, symmetry, and concentration phenomena in polyharmonic nonlocal equations with critical growth.

From a practical perspective, these models correspond to various physical systems displaying long-range interactions (as in quantum mechanics, plasma physics, or thin plate mechanics), and the findings solidify the analytical underpinnings for studying steady states of such systems in higher-order settings.

In theoretical terms, the existence and optimality of extremals for the polyharmonic HLS-type inequalities point toward new geometric and functional-analytic results, potentially impacting the study of critical points for even more general nonlocal functionals.

Future developments might address stability and uniqueness of the nontrivial solutions, the fine structure of blow-up profiles at criticality, and explore extensions to unbounded domains, anisotropic settings, or systems.

Conclusion

This work rigorously establishes the existence of nontrivial solutions for polyharmonic equations with critical Choquard-type nonlinearities and subcritical perturbations. Through variational analysis, sharp threshold computations, and spectral methods, the paper provides the first such results for the general polyharmonic case, extending the analytic theory of nonlocal critical problems and paving the way for deeper qualitative and multiplicity analyses in higher-order, nonlocal nonlinear PDEs (2607.06116).