The effect of boundary geometry in nonlocal critical problems with Hardy-Littlewood-Sobolev exponent

TL;DR

This work analyzes a nonlocal Choquard equation with upper Hardy-Littlewood-Sobolev exponent under mixed Dirichlet-Neumann boundary conditions on a bounded domain. The authors establish how the geometry of the Neumann boundary portion $\Gamma_1$ influences the existence of ground state solutions by introducing a local flatness condition $(\mathbf{H_1})$ and deriving variational estimates. Using an Aubin-type minimization argument and carefully crafted test functions derived from HL ground-state profiles, they show that the minimization constant $S_{HL}(\Gamma_0)$ is attained under either $\beta<3$ (no extra $V$ assumptions) or $V\le0$ near the boundary (any $\beta$ up to $n-1$). This links boundary geometry, particularly mean curvature in the local model, to the solvability of critical nonlocal problems on bounded domains. The results extend prior flatness/mean-curvature conditions for Neumann problems to the nonlocal Choquard setting and provide explicit mechanisms to ensure ground states exist.

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.

Figures (1)

• Figure 1: Domain configuration for Corollary \ref{['C01']}

Theorems & Definitions (17)

• Remark 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Lemma 2.1: lieb2001analysis
• Lemma 3.1
• proof
• Lemma 3.2
• proof