Singular reaction diffusion equations where a parameter influences the reaction term and the boundary conditions - Fonseka, Nalin; Muthunayake, Amila; Shivaji, Ratnasingham; Son, Byungjae (Topological Methods in Nonlinear Analysis, 2021)
We analyse positive solutions to the steady state reaction diffusion equation: equation* 1.11 cases -u’’=λ h(t) f(u) in (0,1), -du’(0)+μ(λ) u(0)=0, u’(1)+μ(λ) u(1)=0, cases equation* where λ> 0 is a parameter, d≥ 0 is a constant, f C^2([0,∞),R) is an increasing function which is sublinear at infinity (_s → ∞f(s)/s=0), h C^1((0,1],(0,∞)) is a nonincreasing function with h_1:=h(1)> 0 and there exist constants d_0> 0, α [0,1) such that h(t)≤ d_0/t^α for all t (0,1], and μ C([0,∞),[0,∞)) is an increasing function such that μ(0)≥ 0. We consider three cases of f, namely, f(0)=0, f(0)> 0 and f(0)< 0. We will discuss existence and multiplicity results via the method of sub-supersolutions. Further, we will establish uniqueness results for λ≈ 0 and λ1.