Sharp point-wise behavior of the positive solutions of a class of degenerate non-local elliptic BVP’s

This paper investigates the degenerate non-local boundary value problem of logistic type \( \left\{ \begin{array}{l} -\Delta u = \lambda u - b(x)u^p - a(x) u\displaystyle \int _\Omega c(y)|u(y)|^r\,dy \quad \hbox {in}\;\; \Omega , \\ u>0 \quad \text{ in }\;\; \Omega ,\ \ u=0 \quad \text{ on }\;\; \partial \Omega , \end{array} \right. \) $$\left\{\begin{array}{c}{\displaystyle -\mathrm{\Delta }u=\lambda u-b(x)u^p-a(x)u{\int }_{\mathrm{\Omega }}c(y){|u(y)|}^{r}dy\text{in}\mathrm{\Omega },}\hfill \\ u>0\text{in}\mathrm{\Omega },u=0\text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)$$ where \(\Omega \subset {\mathbb {R}}^N\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ , \(N\ge 1\) $N\ge 1$ , is a bounded domain with smooth boundary, \(\lambda \in {\mathbb {R}}\) $\lambda \in \mathbb{R}$ is a bifurcation parameter, \(p>1\) $p>1$ , \(r \ge 1\) $r\ge 1$ , a, \(b \in {\mathcal {C}}^\nu ({\overline{\Omega }})\) $b\in {\mathcal{C}}^{\nu }(\overline{\mathrm{\Omega }})$ , \(\nu \in (0,1]\) $\nu \in (0,1]$ , vanish on some subsets of \(\Omega \) $\mathrm{\Omega }$ with positive measure, and \(0<c\in L^\infty (\Omega )\) $0<c\in L^{\infty }(\mathrm{\Omega })$ . The presence of the non-local term prevents us from using the classical sub and supersolutions methods to characterize the existence of positive solutions and ascertain their point-wise behavior with respect to \(\lambda \) $\lambda$ . Combining some ideas going back to Li et al. (Calc Var Partial Differ Equ 60:36, 2021) with the theory of large solutions of as reported by López-Gómez (Metasolutions of parabolic equations in population dynamics, CRCPress, Boca Raton, 2016), we can conduct a detailed study of the point-wise \(\lambda \) $\lambda$ -limit of the set of positive solutions, revealing a behavior substantially different from the one exhibited by the positive solutions of the underlying local problem, because of the presence of the non-local term.