Analysis of reaction–diffusion systems where a parameter influences both the reaction terms as well as the boundary

Abstract

We study positive solutions to steady-state reaction–diffusion models of the form $$ \textstyle\begin{cases} -\Delta u=\lambda f(v);\quad\Omega, \\ -\Delta v=\lambda g(u);\quad\Omega, \\ \frac{\partial u}{\partial \eta }+\sqrt{\lambda } u=0;\quad \partial \Omega, \\ \frac{\partial v}{\partial \eta }+\sqrt{\lambda }v=0; \quad\partial \Omega, \end{cases} $$ { − Δ u = λ f ( v ) ; Ω , − Δ v = λ g ( u ) ; Ω , ∂ u ∂ η + λ u = 0 ; ∂ Ω , ∂ v ∂ η + λ v = 0 ; ∂ Ω , where $\lambda >0$ λ > 0 is a positive parameter, Ω is a bounded domain in $\mathbb{R}^{N}$ R N $(N > 1)$ ( N > 1 ) with smooth boundary ∂Ω, or $\Omega =(0,1)$ Ω = ( 0 , 1 ) , $\frac{\partial z}{\partial \eta }$ ∂ z ∂ η is the outward normal derivative of z. We assume that f and g are continuous increasing functions such that $f(0) = 0 = g(0)$ f ( 0 ) = 0 = g ( 0 ) and $\lim_{s \rightarrow \infty } \frac{f(Mg(s))}{s} = 0$ lim s → ∞ f ( M g ( s ) ) s = 0 for all $M>0$ M > 0 . In particular, we extend the results for the single equation case discussed in (Fonseka et al. in J. Math. Anal. Appl. 476(2):480-494, 2019) to the above system.

$ \Sigma $-shaped bifurcation curves for classes of elliptic systems

“…one can show the existence of three positive solutions for λ > A 1 as in Theorem 1.1(b) and Corollary 1 (see [2]). Next, we discuss the other multiplicity results in Theorem 1.1(a), Theorem 1.2, and Corollary 1.…”

“…Introduction. Recently, there has been growing interest on the study of the reaction diffusion models where a parameter influences the equation as well as the boundary condition (see [1], [2], [6], [8], [7], and [9]). In this paper, we study reaction diffusion systems of the form:…”