发表机构
Center for Mathematics and Interdisciplinary Sciences, Fudan University; Shanghai Institute for Mathematics and Interdisciplinary Sciences（复旦大学数学与交叉科学中心; 上海数学与交叉科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明整数格图（维数≥3）与半空间（维数≥2）上带Hénon位势的离散方程在非线性指数足够大时正解存在，方法结合加权Banach空间、Green核估计与Schauder不动点定理。
AI 中文摘要
我们研究了整数格图与半空间上离散Hénon方程的正解，在半空间情形下带有零Dirichlet边界条件。非线性项由具有正指数的Hénon型位势控制。对于足够大的非线性指数，我们证明了在维数至少为三的整个格图以及维数至少为二的半空间上正解的存在性。由于Hénon位势的存在，标准的变分紧致性无法用于获得正解。我们的证明结合了加权Banach空间、双侧Green核估计以及Schauder不动点定理。
英文摘要
We study positive solutions of a discrete Hénon equation on the integer lattice graph and on the half-space, with zero Dirichlet boundary conditions in the half-space case. The nonlinearity is governed by a Hénon-type potential with positive exponent. For sufficiently large nonlinearity exponents, we prove the existence of positive solutions both on the whole lattice in dimensions at least three and on the half-space in dimensions at least two. Thanks to the Hénon potential, the standard variational compactness is unavailable to obtain the positive solution. Our proofs combine weighted Banach spaces, two-sided Green kernel estimates and the Schauder fixed point theorem.
