An existence result in annular regions times conical shells and its application to nonlinear Poisson systems
Gennaro Infante, Giovanni Mascali, Jorge Rodríguez-López
TL;DR
The work develops a general fixed point index framework for abstract nonlinear operator systems in normed spaces, enabling the localization of coexistence fixed points inside the product of an annular region and a conical shell. By combining star-convex/ wedge theory with a multiplicativity argument and retractions, the authors derive index formulas that ensure existence of fixed points in $K_{r_1,R_1}\times A_{r_2,R_2}$ and provide a constructive pathway for numerical approximation. The theory is applied to elliptic systems, notably nonlinear Poisson problems with Dirichlet data and sign-changing nonlinearities, yielding explicit a priori bounds and a nontrivial, localized solution. An explicit example in a disk and an application to a reaction–diffusion Lotka–Volterra system illustrate both the practical attainability and the effectiveness of parameter tuning to obtain solutions within prescribed ranges.
Abstract
We provide a new existence result for abstract nonlinear operator systems in normed spaces, by means of topological methods. The solution is located within the product of annular regions and conical shells. The theoretical result possesses a wide range of applicability, which, for concreteness, we illustrate in the context of systems of nonlinear Poisson equations subject to homogeneous Dirichlet boundary conditions. For the latter problem we obtain existence and localization of solutions having all components nontrivial. This is also illustrated with an explicit example in which we also furnish a numerically approximated solution, consistent with the theoretical results. We conclude with an application of our results to a reaction--diffusion Lotka--Volterra system with source terms for competing species.