Volume 42_3-4_(2021)_227-251

Multiple positive solutions for some local and non-local elliptic systems arising in desertification models
Jesús Ildefonso Díaz, Jesús Hernández

Abstract. We consider a nonlinear elliptic system proposed in 2007 by E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, in desertification studies. The system models the mutual interaction between the biomass b, the soil-water content w and the surface-water height h. The interactions with the plant environment may lead to some non-local terms which can be approximated by suitable local expressions. Various kinds of feedback processes arise. The change in environmental conditions can be simulated by the change of suitable parameters in the differential equations. Here we consider the case of Dirichlet boundary conditions. After describing some positive solutions corresponding to special values of the parameters, we prove the existence of positive solutions for the local and non-local system. We obtain some bifurcation diagrams showing, rigorously, its starting value and characterizing the supercritical (resp. sub-critical) nature of the branch (something unnoticed before in the previous literature) according to a suitable parameters balance expression. Finally, we prove that if the precipitation datum p(x) grows near the boundary of the domain ∂Ω as d(x,∂Ω)2 then h(x) grows, at most, as d(x,∂Ω)4

Rend. Mat. Appl. (7) 42 (2021) 227-251;  pdf