Characterizing the breakdown of the maximum principle in a class of nonlocal elliptic problems
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.