Abstract
We consider the boundary value problem −u″(x) = λf(u(x)), x ∈ (0, 1); u′(0) = 0; u′(1) + αu(1) = 0, where α > 0, λ > 0 are parameters and f ∈ c2[0, ∞) such that f(0) < 0. In this paper, we study for the two cases ρ = 0 and ρ = θ (ρ is the value of the solution at x = 0 and θ is such that F(θ) = 0 where ) the relation between λ and the number of interior critical points of the nonnegative solutions of the above system.
Highlights
Where α > 0, λ > 0 are parameters, f ∈ c2[0, ∞) and f (0) < 0, and we will assume that there exist β, θ > 0 such that f (s) < 0 on [0, β), f (β) = 0, f (s) ≥ 0, f (s) > 0, lims→∞(f (s)/s) = ∞, and F (θ) = 0 where F (s) =
From (4.15) and (4.10), we have λ2n,2 − λ2n,1 < λ2n+1,2 − λ2n+1,1 the proof is complete
Summary
We consider the two point boundary value problem with NeumannRobin boundary conditions. Where α > 0, λ > 0 are parameters, f ∈ c2[0, ∞) and f (0) < 0, and we will assume that there exist β, θ > 0 such that f (s) < 0 on [0, β), f (β) = 0, f (s) ≥ 0, f (s) > 0, lims→∞(f (s)/s) = ∞, and F (θ) = 0 where F (s) =. Α ∈ (0, ∞), ρ = θ (ρ = 0), (1.1), (1.2), and (1.3) have exactly two nonnegative solutions u2n,i(u2n+1,i), i = 1, 2 with 2n (and 2n+1) interior critical points. It is shown in [3, Theorem 1.4] that for the following
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