Positive solutions for a class of semipositone periodic boundary value problems via bifurcation theory

In this paper, we are concerned with the existence of positive solutions of nonlinear periodic boundary value problems like − u′′ + q(x)u = λ f (x, u), x ∈ (0, 2π), u(0) = u(2π), u′(0) = u′(2π), where q ∈ C([0, 2π], [0, ∞)) with q 6≡ 0, f ∈ C([0, 2π]×R+, R), λ > 0 is the bifurcation parameter. By using bifurcation theory, we deal with both asymptotically linear, superlinear as well as sublinear problems and show that there exists a global branch of solutions emanating from infinity. Furthermore, we proved that for λ near the bifurcation value, solutions of large norm are indeed positive.

Corresponding author.Email: mary@nwnu.edu.cnSemipositone problems arise in many different areas of applied mathematics and physics, such as the buckling of mechanical systems, the design of suspension bridges, chemical reactions, and population models with harvesting effort; see [1,10,17].
Existence of positive solutions for nonlinear second order Dirichlet problems in semipositone case was initially studied by Castro and Shivaji in [4].Henceforth, the existence, multiplicity, and the global behavior of positive solutions of nonlinear second order Dirichlet problems/Robin problems in the semipositone case have been extensively studied by using the method of lower and upper solutions, fixed point theorem in cones as well as the bifurcation theory, see [2,12,16,18] and the references therein.
For nonlinear periodic boundary value problem (1.1), the existence, multiplicity and global behavior of positive solutions have been investigated by several authors via fixed point theorem in cones and the bifurcation theory, one may see J. R. Graef et al. [11], P. J. Torres [19] and Ma et al. [15,20].In particular, the authors of [15,20] showed that there exists an unbounded continuum C emanating from (µ 1 , 0), consisting of positive solutions of (1.1) in the positone case, where µ 1 is the first positive eigenvalue of the linear problem corresponding to (1.1).However, in the semipositone case, (1.1) has no positive solutions for λ large.Let us point out that this is in contrast with the positone case.
It is the purpose of this paper to study the global behavior of positive solutions of (1.1) in semipositone case via bifurcation theory.We shall handle the semipositone problems in which nonlinearities are asymptotically linear, superlinear as well as sublinear at infinity.
After some notation and preliminaries listed in Section 2, we deal in Section 3 with asymptotically linear problems and use bifurcation theory to prove an existence result in the frame of semipositone problems.In Section 4 we discuss superlinear problems, we show that (1.1) possesses positive solutions for 0 < λ < λ * .Similar arguments can be used in the sublinear case, discussed in Section 5, to show that (1.1) has positive solutions provided λ is large enough.

Notation and preliminaries
We denote the usual norm in L r (0, 2π) by • r and the inner product in L 2 (0, 2π) by •, • .We will work in the Banach space Define the linear operator Then L is a closed operator with compact resolvent, and 0 ∈ ρ(L).
We denote by G(x, s) the Green's function associated with the following problem From the Theorem 2.5 of [3], we know that G(x, s) > 0, ∀x, s ∈ [0, 2π] and the solution of the above problem is given by Now, by the positivity of G(x, s) and h(s), we have that u Obviously, 0 < m < M, and 0 < σ < 1.Let K : X → X denote the Green operator of L with periodic boundary conditions, i.e. u = Kv if and only if x ∈ (0, 2π), With the above notation, problem (1.1) is equivalent to Hereafter we will use the same symbol to denote both the function and the associated Nemitskii operator.We say that λ ∞ is a bifurcation from infinity for (2.4) if there exist µ n → λ ∞ and u n ∈ X, such that u n − µ n K f (u n ) = 0 and u n → ∞.Extending the preceding definition, we will say that λ ∞ = +∞ is a bifurcation from infinity for (2.4) if solutions (µ n , u n ) of (2.4) exist with µ n → +∞ and u n → ∞.This is the case we will meet in Section 5.
In the following, we shall apply the Leray-Schauder degree theory, mainly to the mapping Φ λ : X → X.
For R > 0, let B R = {u ∈ X : u < R}, let deg(Φ λ (u), B R , 0) denote the degree of Φ λ on B R with respect to 0 and let i(T, U, X) is the fixed point index of T on U with respect to X.

Asymptotically linear problems
In this section, we suppose that f ∈ C([0, 2π] × R + , R) satisfies (F1) and (F2) there exists m > 0 such that Let λ ∞ = λ 1 m and define Our main result is the following.
The proof of Theorem 3.1 will be carried out in several steps.First of all, we extend f (x, •) to all R by setting Obviously, any u > 0 such that Ψ(λ, u) = 0 is a positive solution of (1.1).
Next, we give two lemmas which will be used later.
Proof.Suppose to the contrary that there exist We may assume µ n → µ > 0, µ = λ ∞ .Set w n := u n u n −1 , we get On the other hand, u n −1 F(u n ) is bounded in X, {w n } is a relatively compact set in X by the compactness of K. Suppose w n → w in X.Then w = 1 and satisfies (i) Assume a > 0. Then the assertion of Lemma 3.
Proof.We prove statement (i); (ii) follows similarly.By Lemma 3.3, the assertion holds for any interval Λ = [λ ∞ + , β], > 0. Suppose now there exist sequences {u n } in X and as in the proof of Lemma 3.3, we conclude that w n → w in X with w > 0. Thus, there exists β > 0 such that w = βφ 1 .Then one has u n = u n w n → +∞ for all x ∈ [0, 2π] and Since λ n > λ ∞ and u n , φ 1 > 0 for n large enough, we infer that f (x, u n ) − mu n , φ 1 < 0 for n large enough and the Fatou lemma yields a contradiction if a > 0.
Proof.Let w 0 be the unique solution of the problem − w + q(x)w = −k(x), a.e. in (0, 2π), Then Set y = u − w 0 .Then − y + q(x)y ≥ 0, a.e. in (0, 2π), and accordingly Proof.Let us assume that for some sequence {u n } in X with u n → ∞ and numbers Then and since F(x, u) ≈ m|u| as |u| → ∞, and Note that u n ∈ D(L) has a unique decomposition where By (F2), there exists M 0 > 0, such that From u n → ∞ and Lemma 3.5, we know that there exits N * > 0, such that and consequently Applying (3.3), it follows that Thus This is a contradiction.
In order to investigate the bifurcation from infinity, we follow the standard pattern and perform the change of variable z = u u −2 (u = 0).Letting one has that λ ∞ is a bifurcation from infinity for (2.4) if and only if it is a bifurcation from the trivial solution z = 0 for Φ = 0. From Lemmas 3.3 and 3.4 it follows by homotopy that Similarly, by Lemma 3.6 one infers, for all τ ∈ [0, 1] and for all λ > λ ∞ , Let us set From (3.4) and (3.5) and the preceding discussion we deduce Lemma 3.7.λ ∞ is a bifurcation from infinity for (2.4).More precisely there exists an unbounded closed connected set Σ ∞ ⊂ Σ that bifurcates from infinity.Moreover, Σ ∞ bifurcates to the left (to the right) provided a > 0 (respectively A < 0).
Proof of Theorem 3.1.By the above lemmas, it suffices to show that if µ n → λ ∞ and u n → ∞ then u n > 0 in [0, 2π] for n large enough.Setting w n = u n u n −1 and using the preceding arguments, we find that, up to subsequence, w n → w in X, and w = βφ 1 , β > 0.Then, it follows that u n > 0 in [0, 2π], for n large enough.
Example 3.9.Let us consider the second-order periodic boundary value problem where Let λ 1 be the first positive eigenvalue corresponding to the linear problem where h(•) ∈ C([0, 2π]) with h ≡ 0. Let φ be the positive eigenfunction corresponding to λ 1 .
Next, we will check that all of conditions in Theorem 3.1 are fulfilled.
The following well-known result of the fixed point index is crucial in our arguments.

Lemma 4.2 ([8]
).Let E be a Banach space and K a cone in E. For r > 0, define K r = {v ∈ K : x < r}.Assume that T : Kr → K is completely continuous such that Tx = x for x ∈ ∂K r = {v ∈ K : x = r}.
(i) If Tx ≥ x for x ∈ ∂K r , then i(T, K r , K)=0.
(ii) If Tx ≤ x for x ∈ ∂K r , then i(T, K r , K)=1.
Proof of Theorem 4.1.As before we set For the remainder of the proof, we will omit the dependence with respect to x ∈ [0, 2π].
We can extend F to γ = 0 by setting F(0, w) = b|w| p and, by (F3), such an extension is continuous.We set Let us point out explicitly that S(γ, •) •) is compact.For γ = 0, solutions of S 0 (w) := S(0, w) = 0 are nothing but solutions of Now, we claim that there exist two constants r 1 , R 1 with 0 < r 1 < R 1 , such that In order to prove (4.4), (4.5) and (4.6), we divide the proof into two steps.
Assume to the contrary that there exists a sequence {w n } of solutions of (4.3) satisfying From the Sturm comparison theorem [13,Theorem 2.6] or the special case of [7, Lemma 5.1] when p = 2, we have w n must change its sign in [0, 2π].This contradicts the fact that w n > 0 on [0, 2π].
Assume on the contrary that (4.5) is not true.Then there exists a sequence w n of solutions of (4.3) satisfying Let v n = w n / w n .From (4.3), we have that is, So lim n→∞ v n = 0 uniformly but this is a contradiction since v n = 1 for all n ∈ N.
To show (4.6) is valid.Define a cone K in X by where σ is from (2.3).A standard argument can be used to show that K F(0, •) : , we deduce This implies Thus the degree deg(S 0 , K R \ Kr , 0) is well defined.
The remaining arguments are the same as that of Theorem 3 of [9] and we will only give a short sketch. Denote It is easy to verify the following conditions where σ, m are from (2.3).
On the other hand, by (A1) there is a δ > 0 such that 0 ≤ w ≤ δ implies where η > 0 satisfying It is obvious that K F(0, w) = w for w ∈ ∂K r .An application of Lemma 4.2 again shows that Now, the additivity of the fixed point index and (4.8), (4.9) together implies Combining this together with the fact S 0 : Therefore, the claim is proved.
Next we show the following result.
Otherwise, there exists a sequence (γ n , w n ) with γ n → 0, w n ∈ {r, R} and w n = K F(γ n , w n ).
Since K is compact then, up to a subsequence, w n → w and S 0 (w) = 0, w ∈ {r, R}, a contradiction with (4.4) and (4.5).
To prove (ii), we argue again by contradiction.As in the preceding argument, we find a sequence w n ∈ X, with {x ∈ [0, 2π] : w n (x) ≤ 0} = ∅, such that w n → w, w ∈ [r, R] and S 0 (w) = 0; namely, w solves (4.3).From the positivity of Green's function G(x, s) and b|w| p , we have w > 0. Therefore w n > 0 on [0, 2π] for n large enough, a contradiction.

Sublinear problems
In this section, we deal with sublinear f , namely f ∈ C([0, 2π] × R + , R) that satisfy (F1) and We will show that in this case positive solutions of (1.1) branch off from ∞ for λ ∞ = +∞.First, some preliminaries are in order.It is convenient to work on Y = C 1 [0, 2π].Following the same procedure as for the superlinear case, we employ the rescaling w = γu, λ = γ q−1 and use the same notation, with q instead of p and Y instead of X.As before, (λ, u) solves (4.1) if (γ, w) satisfies (4.2).Note that now, since 0 ≤ q < 1, one has that λ → +∞ ⇔ γ → 0.
(5.1) Furthermore, it follows from the special case of Dai et al. [7,Theorem 6.1] when p = 2, we get that has a unique positive solution w 0 with w 0 (5.2) implies that v = w 0 is an eigenfunction corresponding to We set D δ = {w ∈ Y : w − w 0 1 ≤ δ} and extend F to γ = 0 by F0 (w) = F(0, w) := b|w| q .
Proof.First of all, we proved that K F : [0, ∞) × D δ → Y is continuous.If 0 < q < 1 the same arguments used for p > 1 show that K F is continuous.Now we consider a situation where q = 0. Let δ > 0 be such that w > 0 for all w ∈ D δ .Obviously, it suffices to show that KF(γ n , w n ) → K F0 (w) whenever γ n → 0 and w n → w in Y. Since w > 0 then γ −1 n w n → +∞, pointwise in [0, 2π].Notice q = 0 implies that lim u→∞ f (x, u) = b, and accordingly, in the Sobolev space H 2,r , ∀r ≥ 1.A standard argument can be used to show that K F : [0, ∞) × D δ → Y is compact.
Proof.By Lemma 5.1, degree theoretic arguments apply to S(γ, w) = w − K F(γ, w).Moreover, note that S 0 (w) = S(0, w) = w − K F0 (w) is C 1 on D δ and its Fréchet derivative S 0 (w 0 ) is given by In particular, for 0 < q < 1, (5.4) implies that all the characteristic values of I − S 0 (w 0 ) are greater than 1.Since w 0 is the unique positive solution of (5.By continuation, we deduce that there exists a connected subset Γ of solutions of S(γ, w) = 0(γ > 0), such that (0, w 0 ) ∈ Γ.Moreover, by an argument similar to that of Lemma 4.3, we get that there exists γ 0 > 0 such that these solutions are positive provided 0 < γ ≤ γ 0 .By the rescaling λ = γ q−1 , u = w/γ, Γ is transformed into a connected subset Σ ∞ of solutions of (1.1).These solutions are indeed positive for all λ > λ * := γ q−1 0 and, according to (5.1), Σ ∞ bifurcates from infinity for λ ∞ = +∞.Remark 5.3.In general, solutions on Σ ∞ can change sign and the behavior of Σ ∞ depends on the definition of f for u < 0. Let us point out that this is in contrast with the positone case; see, for example, the article [7,15,20].