| Abstract: |
| We study the existence of nonnegative and nontrivial solutions to a semilinear elliptic problem in which the classical diffusion term is replaced by an integral operator defined on a measure space. Problems of this type naturally arise in population dynamics, where the solution represents the density of individuals within a given domain. In particular, we establish existence and nonexistence results depending on the sign of the principal spectral value.
Within this framework, we obtain a partition of the domain into subsets on which the problem can be solved independently. On each of these subsets, the strong maximum principle holds, which allows to find strictly positive solutions.
We also present numerical simulations illustrating the theoretical results. These examples show how different choices of measures can model various situations, including random walk processes associated with Markov chains and scenarios where continuous measures of different dimensions coexist within the same domain. |
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