[论文解读] The effect of boundary geometry in nonlocal critical problems with Hardy-Littlewood-Sobolev exponent

Hichem Chtioui, Tuhina Mukherjee+1 more|arXiv (Cornell University)|Jan 25, 2026

Nonlinear Partial Differential Equations被引用 0

一句话总结

本文分析了诺依曼（Neumann）部分边界几何形状如何影响带有上临界 Hardy-Littlewood-Sobolev 指数的混合 Dirichlet–Neumann Choquard 问题的基态解存在性，采用变分方法。

ABSTRACT

In this paper we consider a mixed Dirichlet-Neumann boundary value problem. lem involving Choquard nonlinearity with upper critical exponent in the sense of Hardy- Littlewood Sobolev inequality. We investigate the effect of the geometry of the boundary part where the Neumann condition is prescribed on the existence problem of ground state solutions.

研究动机与目标

• Investigate the existence of nontrivial ground state solutions for a mixed Dirichlet-Neumann Choquard problem on bounded domains.
• Understand how the local geometry of the Neumann boundary portion affects existence results.
• Develop variational techniques to show ground state existence under geometric conditions on the boundary.

提出的方法

• Set up the Choquard equation with mixed Dirichlet-Neumann boundary conditions on a Lipschitz domain.
• Use the Hardy-Littlewood-Sobolev inequality to define the HLS norm and the energy functional.
• Introduce a local geometric condition on the Neumann boundary and construct suitable test functions.
• Perform asymptotic expansions of test functions centered at a boundary point to compare energy levels.
• Apply Aubin-type minimization and Palais-Smale argument to prove existence of ground states.

实验结果

研究问题

• RQ1 Does the boundary geometry of the Neumann portion guarantee the existence of ground state solutions under the given mixed boundary conditions?
• RQ2 How does the flatness order of the boundary near a point influence the energy minimization and existence results?
• RQ3 What role does the external potential V(x) play in the existence results under the geometric condition (H1)?
• RQ4 Can one obtain ground state solutions for general (β up to n-1) boundary flatness using the proposed variational approach?

主要发现

• Under the boundary geometry condition (H1) with β < 3, the problem admits a ground state solution.
• Under (H1) with any flatness order 1 < β ≤ n-1, ground state solutions exist provided V(x) ≤ 0 in a neighborhood.
• The energy functional can be driven below the first PS level using a carefully crafted test function based on a rescaled Aubin–Lions type bubble.
• The analysis shows the energy functional achieves the Hardy-Littlewood-Sobolev best constant in the domain, yielding a ground state.
• Corollaries provide concrete domain configurations where existence is guaranteed, e.g., a domain with a hole where the Neumann part has positive mean curvature at a point.]
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