A three solution theorem for a singular differential equation with nonlinear boundary conditions - Dhanya, Rajendran; Shivaji, Ratnasingham; Son, Byungjae (Topological Methods in Nonlinear Analysis, 2019)
We study positive solutions to singular boundary value problems of the form: equation* cases -u’’ = h(t) f(u)u^α &for t (0,1), u(0) = 0, u’(1) + c(u(1)) u(1) = 0,cases equation* where 0< α< 1, h(0,1]→(0,∞) is continuous such that h(t)≤ d/t^β for some d> 0 and β[0,1-α) and c[0,∞)→ [0,∞) is continuous such that c(s)s is nondecreasing. We assume that f[0,∞)→(0,∞) is continuously differentiable such that [(f(s)-f(0))/s^α ]+τ s is strictly increasing for some τ≥ 0 for s(0,∞). When there exists a pair of sub-supersolutions (ψ,φ) such that 0≤ ψ≤φ, we first establish a minimal solution u and a maximal solution u in [ψ,φ]. When there exist two pairs of sub-supersolutions (ψ_1,φ_1) and (ψ_2,φ_2) where 0≤ ψ_1 ≤ ψ_2 ≤ φ_1, ψ_1 ≤ φ_2 ≤ φ_1 with ψ_2 ot ≤ φ_2, and ψ_2, φ_2 are not solutions, we next establish the existence of at least three solutions u_1, u_2 and u_3 satisfying u_1[ψ_1,φ_2], u_2[ψ_2,φ_1] and u_3[ψ_1,φ_1]([ψ_1,φ_2][ψ_2,φ_1]).