Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion

We consider the nonlinear eigenvalue problem [D(u)u󸀠]󸀠 + λf(u) = 0, u(t) > 0, t ∈ I fl (0, 1), u(0) = u(1) = 0, where D(u) = uk, f(u) = u2n−k−1 + sin u, and λ > 0 is a bifurcation parameter. Here, n ∈ N and k (0 ≤ k < 2n − 1) are constants. This equation is related to the mathematical model of animal dispersal and invasion, and λ is parameterized by the maximum norm α = ‖uλ‖∞ of the solution uλ associated with λ and is written as λ = λ(α). Sincef(u) contains both power nonlinear term u2n−k−1 and oscillatory term sin u, it seems interesting to investigate how the shape of λ(α) is affected by f(u). The purpose of this paper is to characterize the total shape of λ(α) by n and k. Precisely, we establish three types of shape of λ(α), which seem to be new.


Introduction
This paper is concerned with the following nonlinear eigenvalue problems: [ ( ())  ()  ]  +  ( ()) = 0,  ∈  fl (0, 1) , (1) where () =   , () =  2−−1 + sin , and  > 0 is a bifurcation parameter.Here,  ∈ N and  (0 ≤  < 2 − 1) are constants.Bifurcation problems have a long history and there are so many results concerning the asymptotic properties of bifurcation diagrams.We refer to [1][2][3][4][5][6][7][8] and the references therein.Moreover, bifurcation problems with nonlinear diffusion have been proposed in the field of population biology, and several model equations of logistic type have been considered.We refer to [9] and the references therein.In particular, the case () =   ( > 0) has been derived from a model equation of animal dispersal and invasion in [10,11].In this situation,  is a parameter which represents the habitat size and diffusion rate.On the other hand, there are several papers which treat the asymptotic behavior of oscillatory bifurcation curves.We refer to [7,[12][13][14][15][16][17][18][19] and the references therein.Our equation (1) contains both nonlinear diffusion term and oscillatory nonlinear terms.The purpose of this paper is to find the difference between the structures of bifurcation curves of the equations with only oscillatory term and those with both nonlinear diffusion term and the oscillatory term in (1).To clarify our intention, let  = 2 and  = 2. Then (1) is given as The corresponding equation without nonlinear diffusion is the case  = 0 and  = 1, namely, It should be mentioned that, by using a generalized timemap argument in [9], for any given  > 0, there exists a unique classical solution pair (,   ) of ( 1)-( 3) satisfying  = ‖  ‖ ∞ .Furthermore,  is parameterized by  as  = () and is continuous in  > 0. For (5), the following asymptotic formula for () as  → ∞ has been obtained.
Since (4) includes both the nonlinear diffusion function and oscillatory term, it seems interesting how the nonlinear diffusion functions give effect to the structures of bifurcation curves.Now we state our main results.
The proofs depend on the generalized time-map argument in [9] and stationary phase method (cf.Lemma 4).It should be mentioned that if we apply Lemma 4 to our situation, careful consideration about the regularity of the functions is necessary.
For  ≥ 0, we put It is known from [9] that if (  , ()) ∈  2 () × R + satisfies (1)-( 3), then In what follows, we denote by  various positive constants independent of  ≫ 1.For 0 ≤  ≤ 1 and  ≫ 1, we have By this, (19), and Taylor expansion, we have from [9, (2.5)] that where We see from ( 24) and ( 25) that if we obtain the precise asymptotic formula for () as  → ∞, then we obtain Theorem 2. To do this, we apply the stationary phase method to our situation.By combining [13, Lemma 2] and [7, Lemmas 2.24], we have the following equality.( In particular, by taking the imaginary part of ( 27), as  → ∞, ) . ( We note that, to obtain (27), we have to be careful about the regularity of  and .
By (24) and Lemma 5, we obtain Theorem 2 immediately.Thus the proof is complete.

Proof of Theorem 3
In this section, let 0 <  ≪ 1.The proofs of Theorem 3 (i)-(v) are similar.Therefore, we only prove (i) and (iv).