On the existence and multiplicity of positive solutions to classes of steady state reaction diffusion systems with multiple parameters

2023/07/22 by Shabanpour, A., Rasouli, S. H., Fonseka, N.
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.12000

Abstract

We study positive solutions to the steady state reaction diffusion systems of the form: \-Δu = λf(v)+μh(u), · amp; Ω,
-Δv = λg(u)+μq(v), · amp; Ω,
(∂ u)/(∂ η)+√[]λ+μ u=0, · amp; ∂Ω,
(∂ v)/(∂ η)+√[]λ+μ v=0, · amp; ∂Ω,. where λ,μ>0 are positive parameters, Ω is a bounded in ℝ^N(N>1) with smooth boundary ∂ Ω, or Ω=(0,1), (∂ z)/(∂ η) is the outward normal derivative of z. Here f, g, h, q∈ C² [0,r)∩ C[0,∞) for some r>0. Further, we assume that f, g, h and q are increasing functions such that f(0) = g(0) = h(0) = q(0) = 0, f^′(0), g^′(0), h^′(0), q^′(0) > 0, and lim_{s→ ∞}(f(M g(s)))/(s)=0 for all M>0. Under certain additional assumptions on f, g, h and q we prove our existence and multiplicity results. Our existence and multiplicity results are proved using sub-super solution methods.

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