Multiple Positive Solutions for a Class of Neumann Problems

We study the existence of multiple positive solutions of the Neumann problem −u (x) = λ f (u(x)), x ∈ (0, 1), u (0) = 0 = u (1), where λ is a positive parameter, f ∈ C([0, ∞), R) and for some β > 0 such that f (0) = 0, f (s) > 0 for s ∈ (β, ∞), f (s) < 0 for s ∈ (0, β), and θ (> β) is the unique positive zero of F(s) = s 0 f (t) dt. In particular, we prove that there exist at least 2n + 1 positive solutions for λ ∈ n 2 π 2


Introduction
In this paper, we are concerned with the existence of multiple positive solutions to the Neumann problem −u (x) = λ f (u(x)), x ∈ (0, 1), where λ is a positive parameter, f ∈ C([0, ∞), R) and for some β > 0 such that f (0) = 0, f (s) > 0 for s ∈ (β, +∞), f (s) < 0 for s ∈ (0, β), and θ (> β) is the unique positive zero of F(s) = s 0 f (t) dt.The Neumann problems have played a significant role in mathematical physics (for example, equilibrium problems concerning beams, columns, or strings and so on), and hence have attracted the attention of many researchers over the last two decades, see [3,9,11] and Corresponding author.Email: mary@nwnu.edu.cn the references therein.The existence and multiplicity of positive solutions for the Neumann boundary value problems were investigated in connection with various configurations of f by the fixed point theorems in [3,11] and by a detailed analysis of time-map associated with (1.1) in [9].In [8], Maya and Shivaji obtained multiple positive solutions for a class of semilinear elliptic boundary value problems by using sub-super solutions arguments when f ∈ C 1 satisfies the following conditions: (f4) f is eventually increasing and lim u→∞ f (u) u = 0. Recently, Ma [5] studied the global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems by using global bifurcation techniques.For the other results related to the existence of nodal solutions, see [6,7] and the references therein.
Motivated by the above papers, in this paper, we investigate the existence of multiple positive solutions of (1.1) by applying the bifurcation and continuation methods.In fact, we will transform the Neumann problem into the Dirichlet problem by virtue of the continuation methods, and then we are concerned with determining values of λ, for which there exist nodal solutions of the Dirichlet boundary value problem by means of bifurcation techniques.Consequently, we give the existence results of positive solutions of the problem (1.1), under the following assumptions.
We will establish the following theorem.
Theorem 1.1.Let (H1)-(H3) hold and n ∈ N. Then there exist at least 2n + 1 positive solutions of (1.1) for λ ∈ n 2 π 2 f (β) , ∞ .Now to illustrate Theorem 1.1, let us consider the simple example f (s) = s 2 (s − 1) for s ≥ 0. Hence β = 1 and f (β) = 1.Thus, given n ∈ N, problem (1.1) has at least 2n + 1 positive solutions for all λ ∈ (n 2 π 2 , ∞).Remark 1.2.Compared with the configurations of f in [8,9], we only demand f ∈ C[0, ∞), so that the quadrature technique does not apply to (1.1).Even if f ∈ C 1 satisfies (H1), it seems rather difficult to make a detailed analysis of the so-called time map to trace down the positive solution of (1.1).[8] obtained multiple positive solutions for a class of semilinear elliptic boundary value problems when f satisfies (f1)-(f4).Notice that f ∈ C 1 implies that f is Lipschitz continuous in [0, β], and so (H3) is satisfied.It is worth to be pointed out that in Theorem 1.1 neither f ∈ C 1 , nor a growth condition at infinity is required.

Remark 1.3. Maya and Shivaji
The paper is organized as follows.In Section 2 we introduce some notations and auxiliary results and in Section 3 we prove our main result.

Let
The following results are somewhat scattered in Miciano and Shivaji [9].
Remark 2.3.Any zero of f is a solution of (1.1).
To study positive solution u(x) of (1.1) which has n − 1 interior critical points at k/n, (k = 1, 2, . . ., n − 1), it suffices by Lemma 2.2, to study solution Thus we only need to study the form of positive solution u.

Proof of the main result
Proof of Theorem 1.1.We first prove the case n = 1.It is divided into three steps. Step It is easy to see that the solution u > 0 of (1.1) is equivalent to the solution v of (3.1) with v > −β.
To study solutions of (3.1), we consider the auxiliary problem where 2) is an autonomous equation, we may consider the Dirichlet problem Note that any solution w(x) of (3.4) is symmetric with respect to any point x 0 ∈ (0, 2) such that w (x 0 ) = 0.In order to find a positive solution of (1.1), it is enough to find such solution of (3.4), which have exactly one simple zero in (0, 2) and is negative near x = 0.
According to [1], we extend the function g to a continuous function g defined on R in such a way that g(s) > 0 for all s < −β.In the sequel of the proof we shall replace g with g, however, for the sake of simplicity, the modified function g will still be denoted by g.

Clearly, lim |w|→0
ζ(w) as a bifurcation problem from the trivial solution w = 0. Note that (3.6) is equivalent to (3.5).By the Krasnoselskii-Rabinowitz bifurcation theorem (see [2,Theorem 22.8]), the following result holds.Lemma 3.1.λ k is a bifurcation point of (3.6) and the associated bifurcation branch C k in R × E whose closure contains (λ k , 0) is either unbounded or contains a pair (λ j , 0) and j = k, where Let E = R × E under the product topology.Let S + k denote the set of functions in E which have exactly k − 1 simple zeros in (0, 2) and are positive near t = 0, and set S − k = −S + k , and Further it follows from (3.7) that Proof.Suppose on the contrary, if there exists (λ m , w m ) → (λ j , 0) when m → +∞ with (λ m , w m ) ∈ C k , w m ≡ 0 and j = k.Let y m := w m w m , then y m should be a solution of the problem Proof.Taking into account Lemma 3.1 and Lemma 3.2, we only need to prove that w n , then u n should be a solution of the problem By (3.8), (3.10) and the compactness of L −1 we obtain that for some convenient subsequence u n → u 0 = 0 as n → +∞.Now u 0 verifies the equation  From (H1)'-(H3)', by a proof similar to that of Theorem 2.1 of [5], for any (λ, . Multiplying (3.5) by w (x) and then integrating from x 1 to x 2 , we have From Lemma 3.5 and w ∈ C 1 [0, 2] is bounded, and so C − 2 is unbounded in the direction of λ.

. 9 )Lemma 3 . 3 .
By(3.8),(3.9)and the compactness of L −1 , we obtain that for some convenient subsequence y m → y 0 = 0 as m → +∞.Now y 0 verifies the equation −y 0 = λ j f (β)y 0 and y 0 = 1.Hence y 0 ∈ S j which is an open set in E, and as a consequence for some m large enough, y m ∈ S j , and this is a contradiction.From each (λ k , 0) it bifurcates an unbounded continuum C k of solutions to problems (3.6) with exactly k − 1 simple zeros.