发表机构
Universidad de Sevilla（塞维利亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文表征一类非局部椭圆问题中最大值原理随参数失效的两种退化类型，给出死核解出现的充分条件，并研究最大化临界阈值的优化问题，证明其无界性及极大值存在性。
AI 中文摘要
本文考虑一类含非负参数$a$的非局部椭圆方程，该参数量化非局部项的强度，问题在光滑有界域上设定，并带有齐次Dirichlet边界条件及给定的非负源项$f$。此类问题出现在多个应用中，包括热敏电阻行为的建模及具有拥挤效应的种群动力学。我们关注当参数$a$增大时最大值原理的失效。我们刻画了在临界参数$a^*$处可能发生（可能同时发生）的两种退化类型：解可能在域的内部点消失，可能形成死核区域，或其法向导数可能在边界上消失。特别地，我们获得了死核解出现的充分条件，从而回答了近期文献中提出的一个开放问题。构造了多个一维和二维例子以说明每种可能情形。此外，我们研究了一个受生物学启发的优化问题，其目标是在一类可容许的资源分布上最大化临界阈值$a^*$。对于具有给定总质量、一致上界及内部接触条件的源项，我们证明临界阈值是无界的，且相应的总种群可以任意小。我们还在一类具有给定总种群正下界、源项一致界及相同内部接触条件的类中建立了极大值的存在性。
英文摘要
In this paper, we consider a nonlocal elliptic equation involving a nonnegative parameter $a$ that quantifies the strength of the nonlocal term, posed in a smooth bounded domain with homogeneous Dirichlet boundary conditions and a given nonnegative source term $f$. This class of problems arises in several applications, including the modeling of thermistor behaviour and population dynamics with crowding effects. We focus on the breakdown of the maximum principle as the parameter $a$ increases. We characterize the two types of degeneracy that may occur, possibly simultaneously, at the critical parameter $a^*$: the solution may vanish at an interior point of the domain, possibly forming a dead-core region, or its normal derivative may vanish on the boundary. In particular, we obtain sufficient conditions for the emergence of dead-core solutions, thereby answering an open question raised in the recent literature. Several one- and two-dimensional examples are constructed to illustrate each possible scenario. Additionally, we study an optimization problem motivated by biological considerations, where the goal is to maximize the critical threshold $a^*$ over an admissible class of resource distributions. For sources with prescribed total mass, a uniform upper bound, and an interior-contact condition, we show that the critical threshold is unbounded and that the corresponding total population can be arbitrarily small. We also establish the existence of a maximizer in a class with a prescribed positive lower bound on the total population, a uniform bound on the sources, and the same interior-contact condition.
