Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian

In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight. By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic $p$-Laplacian with the sign-changing weight. In particular, we show that there exist two simple, isolated, principal eigenvalues $\lambda_0^+$ and $\lambda_0^-$. Furthermore, under some natural hypotheses on perturbation function, we show that $(\lambda_0^\nu,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $\mathscr{C}_\nu^{+}$ and $\mathscr{C}_\nu^{-}$, consisting of the continuum $\mathscr{C}_\nu$ emanating from $(\lambda_0^\nu, 0)$, where $\nu\in\{+,-\}$. As an application of the above result, we study the existence of one-sign solutions for a class of quasilinear periodic boundary problems with the sign-changing weight. Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter $\lambda$ are also studied.


Introduction
In the past few decades, periodic boundary value problems have attracted the attention of many specialists in differential equations because of their interesting applications. For example, the application in looking for spatially periodic solutions of the well-known Camassa-Holm equation, see [1,8,9,10,14]. The Camassa-Holm equation is a recently discovered model for the propagation of shallow water waves of moderate amplitude [13,30] and some authors have already indicated recently that the equation might be relevant to the modeling of tsunamis [12,31]. As the recent year examples, we mention the papers of Atici and Guseinov [4], Jiang et al. [29], Li [33], O'Regan and Wang [43], Torres [49], Zhang and Wang [51], Graef et al. [27] and references therein. Their main tool is the fixed-point theorem of cone expansion/compression type. Ma et al. [39,40] studied the existence of positive solutions for the second-order periodic boundary value problems by making use of the bifurcation techniques.
Recently, Dai and Ma [17] established unilateral global bifurcation theory for one-dimensional p-Laplacian problems with 0-Dirichlet boundary condition. Moreover, Dai and Ma [17], Dai [16] also studied the existence of nodal solutions for the one-dimensional p-Laplacian problems based on the unilateral global bifurcation theory. For the abstract unilateral global bifurcation theory, we refer the reader to [17,19,20,36] and the references therein.
Following the above spectrum results, we shall show that (λ ν 0 , 0) is a bifurcation point of one-sign solutions to problem (1.1) and there are two distinct unbounded sub-continua C + ν and C − ν , consisting of the continuum C ν bifurcating from (λ ν 0 , 0), where ν ∈ {+, −}. On the basis of the unilateral global bifurcation result, we investigate the existence of one-sign solutions for the following periodic p-Laplacian problem − (ϕ p (u ′ )) ′ + q(x)ϕ p (u) = λm(x)f (u), 0 < x < T, u(0) = u(T ), u ′ (0) = u ′ (T ), (1.4) where f ∈ C(R), λ is a parameter. Here, we shall establish some results of existence, multiplicity and nonexistence of one-sign solutions for problem (1.4) according to the asymptotic behavior of f at 0 and ∞ and the fact of whether f possesses zeros in R\{0}. Our results extend and improve the corresponding ones of [27]. To the best of our knowledge, most results of this paper are new even in the case of p = 2. We now give a brief description of the contents of the paper. In Section 2, with the aid of the Ljusternik-Schnirelmann theory and operator theory, we study the variational eigenvalues of problem (1.3). Moreover, as a byproduct, we also establish several important properties of a quasilinear operator which itself possesses an independent importance. The results of this section partially extend the corresponding ones of [5,6].
In Section 3, we prove some properties of the principle eigenvalues λ + 0 and λ − 0 . More precisely, we shall show that λ + 0 and λ − 0 are simple, isolated, principal eigenvalues (their corresponding eigenfunctions are positive or negative) and continuous with respect to p. It is well-known that the continuity of λ + 0 and λ − 0 with respect to p is crucial in the studying of the global bifurcation phenomena for p-Laplacian. We use the method established by Del Pino et al. [22,23] to prove this result but with some extra effort since the boundary condition is different from [23]. To the best of our knowledge, this result is new even in the case of m ≥ 0.
In Section 4, we establish the unilateral global bifurcation theory for problem (1.1). In the global bifurcation theory of differential equations, it is well-known that a change of the index of the trivial solution implies the existence of a branch of nontrivial solutions, bifurcating from the set of trivial solutions which is either unbounded or returns to the set of trivial solution. Hence, the index formula of an isolated zero is very important in the study of the bifurcation phenomena for differential equations. Firstly, we establish an index formula for p = 2 by the linear compact operator theory. Then by use of the index formula and the deformation along p, we prove an index formula involving the problem (1.3) which guarantees (λ ν 0 , 0) is a bifurcation point of nontrivial solutions to problem (1.1). Furthermore, by an argument similar to that of [17], we can get unilateral global bifurcation results for problem (1.1).
In Section 5, we study the existence of one-sign solutions for problem (1.4) with signum condition according to the asymptotic behavior of f at 0 and ∞. The results of this section extend and improve the corresponding ones of [ In Section 6, we show a result involving the uniqueness and dependence of solutions on the parameter. This result extends and improves the corresponding ones to [27,Theorem 2.2] even in the case of p = 2. To prove this result, we introduce a new method which is different from that of [27,34,35].
Finally, Section 7 is devoted to study the existence of one-sign solutions for problem (1.4) without signum condition. To do this, following some ideas from [46], we establish a unilateral global bifurcation theorem from infinity for problem (1.1). This theorem, as an independent result, is of interest too. Our results of this section extend and improve the corresponding results of [39].

Variational eigenvalues
In this section, we shall establish the eigenvalue theory for problem (1.3) via the Ljusternik-Schnirelmann theory. Let It is not difficult to verify that W 1,p T (0, T ) is a real Banach space. For simplicity, we write u n ⇀ u and u n → u to indicate the weak convergence and strong convergence of sequence {u n } in W 1,p T (0, T ), respectively.
First, we recall the definition of weak solution.
For the regularity of weak solution, we have the following result.
is also a classical solution of problem (1.3).
In order to prove Lemma 2.1, we need the following technical result.
Proof. The conclusion is a direct corollary of Lagrange mean theorem, we omit the proof here.
Define the functional on W 1,p It is obvious that the functional Φ is continuously Gâteaux differentiable. Denote L := Φ ′ : We have the following properties about the operator L.
Proof. (i) It is not difficult to verify that L is continuous. For any u, v ∈ W 1,p i.e., L is monotone. In fact, L is strictly monotone.
Thus, we obtain If 1 < p < 2, (2.8) implies that u ′ = v ′ and u = v, which is a contradiction. If p ≥ 2, (2.8) implies that u ′ = v ′ and u = v which contradicts u = v in W 1,p T (0, T ) or |u ′ | ≡ 0 ≡ |u|. If the later case occurs, we get v = u ≡ 0, which is a contradiction. Therefore, L(u)−L(v), u−v > 0. It follows that L is a strictly monotone operator on W 1,p T (0, T ). In view of (2.7), u ′ n (u n ) converges in measure to u ′ (u) in (0, T ), so we get a subsequence (which we still denote by u n ) satisfying u ′ n (x) → u ′ (x) and u n (x) → u(x), a.e. x ∈ (0, T ). By Fatou's Lemma we get According to (2.9) and (2.10) we obtain By a similar method to prove [26, Theorem 3.1], we have Therefore, u n → u, i.e., L is of type (S + ).
(iii) It is clear that L is an injection since L is a strictly monotone operator on W 1,p T (0, T ). Since L is coercive, thus L is a surjection in view of Minty-Browder Theorem (see [50,Theorem 26A]). Hence L has an inverse map L −1 : W 1,p T (0, T ) * → W 1,p T (0, T ). Therefore, the continuity of L −1 is sufficient to ensure L to be a homeomorphism. If The coercive property of L implies that {u n } is bounded in W 1,p T (0, T ). We can assume that Since L is of type (S + ), u n k → u 0 . Furthermore, the continuity of L implies that L (u 0 ) = L(u). By injectivity of L, we have u 0 = u. So u n k → u. We claim that u n → u in W 1,p T (0, T ). Otherwise, there would exist a subsequence u m j of {u n } in W 1,p T (0, T ) and an ε 0 > 0, such that for any j ∈ N, we have u m j − u ≥ ε 0 . But reasoning as above, u m j would contain a further subsequence u m j l → u in W 1,p T (0, T ) as l → +∞, which is a contradiction to u m j l − u ≥ ε 0 . Therefore, L −1 is continuous.
Define the functional Ψ : The following theorem is the main result of this section. Moreover, where γ(K) is the genus of K. Then the weak form (also classical form by Lemma 2.1) of problem (1.3) on M can be equivalently written as We claim that Φ satisfies the Palais-Smale condition at any level set c. Suppose that {u n } ⊂ M, |Φ (u n )| ≤ c and Φ ′ (u n ) → 0. Then for any constant θ > p, we get Hence, { u n } is bounded. Up to a subsequence, we may assume that Obviously, Proposition 2.2 (iii) implies that 0 is not the eigenvalue of problem (1.3). Now, applying Corollary 4.1 of [48], we obtain that problem (1.3) possesses a sequence of positive eigenvalues Moreover, we have that In particular, if k = 0, taking K = {u, −u|u ∈ M}, we can get that In the case of λ < 0, we restate eigenvalue problem (1.3) as the following Using the above result, we have that (2.12) possesses a sequence of positive eigenvalues 3) also possesses a sequence of negative eigenvalues Similar to λ + 0 , we also get that This completes the proof.
Remark 2.2. Note that if m ≥ 0 but m ≡ 0, we can only get the positive eigenvalues.

Properties of positive minimal and negative maximal eigenvalues
In this section, we are going to study the properties of λ + 0 and λ − 0 . These properties, such as simplicity, isolation and the continuity with respect to p, are important in the study of the global bifurcation phenomena for p-Laplace problems, see [23,32,44].
Similar to the results of the positive weight [6, Theorem 3.1], we have the following theorem. 4. Any eigenfunction u associated to λ = λ + 0 and λ = λ − 0 changes sign.
Proof. We only consider the case of λ ≥ 0 since the proof of λ < 0 can be given similarly. Using a proof similar to that of [6, Theorem 3.1] with obvious changes, we can obtain the properties of 1, 2 and 3. However, the method which is used to prove Theorem 3.1 (c) of [6] cannot be used directly here to prove 4 because m is a sign-changing function.
Suppose on the contrary that λ > λ + 0 and there exists an eigenfunction u ≥ 0, i.e., (λ, u) satisfies problem (1.3). Similar to the proof of [6, Theorem 3.1], we can show that u > 0 on [0, T ]. Multiplying the first equation of problem (1.3) by u, we obtain after integration by parts So by scaling we may suppose that Let u + 0 be the eigenfunction corresponding to λ + 0 satisfying This is a contradiction. Theorem 3.1 has shown that λ + 0 is left-isolated and λ − 0 is right-isolated. Furthermore, we can show that λ + 0 and λ − 0 are isolated as the following.
Proof. We only prove the isolated property of λ + 0 since the case λ − 0 is completely analogous. Assume by contradiction that there exists a sequence of eigenvalues λ n ∈ λ + 0 , δ + which converges to λ + 0 . Let u n be the corresponding eigenfunctions. Theorem 3.1 implies that u n changes sign. Integration by parts helps to yield On the other hand, The above inequality and the variational characterization of λ + 0 imply that Then Theorem 3.1 follows that v is positive or negative. Without loss of generality, we may as- Thus u n ≥ 0 for n large enough. This contradicts u n changing sign.
It is easy to see from Theorem 2.1 that the values of λ + 0 and λ − 0 are dependent on p. Hence, we can rewrite λ + 0 and λ − 0 as λ + 0 (p) and λ − 0 (p) to indicate this dependence. In fact, we can describe this relation more precisely as the following proposition does, and this proposition is crucial to prove our main results in this paper. In order to prove this proposition, we need the following results.
Proof. We define another norm on W 1,p T (0, T ) by It is easy to verify that · * is equivalent to · . From now on, we use W 1,p T,0 (0, T ) to denote the space W 1,p T (0, T ) with the norm · * . In [41], Mawhin and Willem gave another definition of weak derivative which called T -weak derivative by Fan and Fan [25]. Letu denote the T -weak derivative of u ∈ L 1 (0, T ). Define Proof. From the variational characterization of λ + 0 (p) it follows that Otherwise, for all n ∈ N, we have The fact that X ֒→ C[0, T ] is compact and u n → u in X imply that (3.4) and (3.5) imply that u = cu + 0 . This is a contradiction. Thus, (3.2) and (3.3) implies that λ + 0 (p) ≤ λ ∞ 0 (p). Therefore, we have Proof of Proposition 3.2. We only show that λ + 0 : (1, +∞) → R is continuous since the proof that λ − 0 is similar. In the following proof, we shall simply write λ + 0 as λ 0 . Lemma 3.2 has shown that To do this, let u ∈ C ∞ T (R). Then, from (3.6), we have that Applying the Dominated Convergence Theorem we find lim sup Relation (3.8), the fact that u is arbitrary and (3.6) yield lim sup Thus, to prove (3.7) it suffices to show that lim inf Let us fix ε 0 > 0 so that p − ε 0 > 1 and for each 0 For 0 < ε < ε 0 and k large enough, (3.10) and Hölder's inequality imply that On the other hand, we also have Clearly, (3.12) and (3.13) show that {u k } is a bounded sequence in W 1,p−ε Passing to a subsequence if necessary, we can assume that u k ⇀ u in W 1,p−ε (3.14) We note that (3.11) implies that for all k ∈ N. Thus letting k → +∞ in (3.15) and using (3.14), we find lim inf On the other hand, since u k ⇀ u in W 1,p−ε T (0, T ), from (3.12) we obtain that Hence u ∈ W 1,p (0, T ). While we know that u ∈ W 1,p−ε T (0, T ) for each 0 < ε < ε 0 . It follows that u(0) = u(T ). Thus, we obtain that u ∈ W 1,p T (0, T ). Finally, combining (3.16) and (3.17) we obtain lim inf This relationship together with the variational characterization of λ 0 (p) implies (3.9) and hence (3.7). This concludes the proof of the proposition.

Unilateral global bifurcation
From now on, we use X to denote the space W 1,p T (0, T ). We start this section by considering the following auxiliary problem for a given h ∈ X * .
Lemma 4.1. If h ∈ X * , then problem (4.1) has a unique weak solution.
Proof. For any v ∈ X, we define h, v := T 0 hv dx. It is easy to verify that h is a continuous linear functional on X. Since L is a homeomorphism, (4.1) has a unique solution.
Let G p (h) denote the unique solution to problem (4.1) for a given h ∈ X * . Proposition 2.2 implies that G p : X * → X is continuous. Since X embeds compactly into L q (0, T ) for each q ∈ [1, +∞] it follows that the restriction of G p to L q ′ (0, T ) is a completely continuous operator, where q ′ = q/(q − 1) (+∞) if q > 1 (q = 1). Define T p λ (u) = G p (F (λ, u)) on X, where F (λ, u) denotes the usual Nemitsky operator associated to λm(x)ϕ p (u(x)). The compact embedding of X ֒→ L p (0, T ) and Theorem 1.7 of [3] imply that T p λ : X → L p ′ (0, T ) is completely continuous. Thus, T p λ : X → X is completely continuous. Let Ψ p,λ defined on X be given by Ψ p,λ (u) = u − T p λ (u).
where δ + 2 and δ − 2 are chosen in such a way that there is no other eigenvalue in Proof. We divide the proof into two cases. Case 1. λ ≥ 0.
In this case, we consider a new sign-changing eigenvalue problem It is easy to check that Thus, we may use the result obtained in Case 1 to deduce the desired result.
As far as the general p, we can compute it through the deformation along p.
Define the Nemitskii operator H λ : X → L 1 (0, T ) by Then it is clear that H λ is continuous operator and problem (1.1) can be equivalently written as Using a similar method to prove [17, Theorem 2.1] with obvious changes, we may obtain the following result. Remark 4.1. It is not difficult to verify that the conclusion of Lemma 2.1 is also valid for problem (1.1). It follows that u is also a classical solution for any (λ, u) ∈ C ν .
Next, we shall prove that the first choice of the alternative of Theorem 4.1 is the only possibility. Let P + denote the set of functions in X which are positive in [0, T ]. Set P − = −P + and P = P + ∪ P − . It is clear that P + and P − are disjoint and open in X. Finally, let K ± = R × P ± and K = R × P under the product topology.
By an argument similar to that of [17, Theorem 2.1], we obtain that for some convenient subsequence, v n → v 0 as n → +∞. It is easy to see that λ, v 0 verifies problem (1.3) and v 0 = 1.
On the other hand, we can easily show that the bifurcation points must be eigenvalues. Thus, C ν does not join to (0,0) because 0 is not the eigenvalue of problem (1.3). Clearly, Proposition 2.2 implies that (0,0) is the only solution of problem (1.1) for λ = 0. Hence, we have C ν ∩ ({0} × X) = ∅. It follows that λ = λ −ν 0 . Theorem 3.1 follows v 0 must change its sign, and as a consequence for some n large enough, u n must change sign. This is a contradiction.
Proof. If the result doesn't hold, then there would be a sequence {(λ n , u n )} ∈ C ν ∩ O such that u n ≡ 0, u n ∈ K and (λ n , u n ) → (λ ν 0 , 0). Let v n := u n / u n , then v n should be the solutions of the problem By an argument similar to that of Lemma 4.4, we obtain for some convenient subsequence, v n → v 0 as n → +∞. It is easy to see that (λ ν 0 , v 0 ) verifies problem (1.3) and v 0 = 1. Then Theorem 3.1 implies that v 0 is positive or negative. Without loss of generality, we may assume that v 0 > 0 on [0, T ]. This is impossible since K is open.
Furthermore, applying the similar method to prove [36, Lemma 6.4.1] with obvious changes, we may obtain the following result, which localizes the possible solutions of (1.1) bifurcating from (λ ν 0 , 0).
By an argument similar to prove [17,Theorem 3.2] with obvious changes, we may obtain the following unilateral global bifurcation result. Theorem 4.3. There are two distinct unbounded sub-continua of solutions to problem (1.1), C + ν and C − ν , consisting of the bifurcation branch C ν and Moreover, if we pose more strict assumption on g as the following: Then we have We consider the following problem Applying Theorem 4.3 to problem (4.2), we obtain that there are two distinct unbounded subcontinua of solutions to problem (4.2), C + ν and C − ν , consisting of the bifurcation branch C ν and ν is also the solution branch of problem (1.1).

Remark 4.2.
Note that if m ≥ 0 but m ≡ 0, we can only get the component C σ + emanating from λ + 0 , 0 . Thus, our results in this section are new even in the definite weight case.

One-sign solutions with signum condition
In this section, we shall investigate the existence and multiplicity of one-sign solutions to problem (1.4). Let Through out this section, we always suppose that f satisfies the following signum condition Clearly, (A2) implies f (0) = 0. Hence, u = 0 is always the solution of problem (1.4). Applying Theorem 4.3, we shall establish the existence of one-sign solutions of problem (1.4) as the following.
In order to prove Theorem 5.1, we need the following Sturm-type comparison result.
Also let u 1 , u 2 be solutions of the following differential equations Let y n ∈ X be a solution of the equation Then the number of zeros of y n | I goes to infinity as n → +∞.
Proof. Taking a subsequence if necessary, we may assume that as j → +∞, where λ j is the jth eigenvalue of the following problem ) and q(t) = q (T /(b − a)(t − a)). By some simple computations, we can show Let ϕ j be the corresponding eigenvalue of λ j . Theorem 3.1 of [47] implies that the number of zeros of ϕ j I goes to infinity as j → +∞. By Lemma 5.1, one obtains that the number of zeros of y n | I goes to infinity as n → +∞. It follows the desired results.
Step 1. We show that there exists a constant M such that λ n ∈ (0, M] for n ∈ N large enough.
On the contrary, we suppose that lim n→+∞ λ n = +∞. We note that The signum condition (A2) implies that there exists a positive constant ̺ such that f n (x) ≥ ̺ for any x ∈ [0, T ]. By Lemma 5.2, we get that u n must change its sign in [0, T ] for n large enough, and this contradicts the fact that u n ∈ C σ + .
Step 2. We show that C σ + joins λ + 0 /f 0 , 0 to λ + 0 /f ∞ , +∞ . It follows from Step 1 that u n → +∞. Let ξ ∈ C(R) be such that f (s) = f ∞ ϕ p (s) + ξ(s). Then lim |s|→+∞ ξ(s)/ϕ p (s) = 0. Let ξ(u) = max u≤|s|≤2u |ξ(s)|. Then ξ is nondecreasing and We divide the equation by u n and set u n = u n / u n . Since u n are bounded in X, after taking a subsequence if necessary, we have that u n ⇀ u for some u ∈ X. Moreover, from (5.3) and the fact that ξ is nondecreasing, we have that where · ∞ denotes the usual norm of C[0, T ] and C 0 is the embedding constant of X ֒→ C[0, T ]. By the continuity and compactness of F λ , it follows that where λ = lim n→+∞ λ n , again choosing a subsequence and relabeling it if necessary.
Proof. We shall only prove the case λ > 0 since the proof for the other case is completely analogous. In view of Theorem 5.1, we only need to show that C σ + joins λ + 0 /f 0 , 0 to (+∞, +∞). Suppose on the contrary that there exists λ M be a blow up point and λ M < +∞. Then there exists a sequence {λ n , u n } such that lim By the compactness of F λ , we obtain that for some convenient subsequence v n → v 0 as n → +∞. Letting n → +∞, we obtain that v 0 ≡ 0. This contradicts v 0 = 1.
Next, we shall need the following topological lemma.
Lemma 5.3 (see [38]. Let X be a Banach space and let C n be a family of closed connected subsets of X. Assume that: (i) there exist z n ∈ C n , n = 1, 2, . . ., and z * ∈ X, such that z n → z * ; (ii) r n = sup x x ∈ C n = +∞; (iii) for every R > 0, ∪ +∞ n=1 C n ∩ B R is a relatively compact set of X, where Then there exists an unbounded component C of D =: lim sup n→+∞ C n and z ∈ C.
Proof. Inspired by the idea of [2], we define the cut-off function of f as the following We consider the following problem Clearly, we can see that lim n→+∞ f n (s) = f (s), (f n ) 0 = f 0 and (f n ) ∞ = n. Theorem 5.1 implies that there exists a sequence of unbounded continua (C σ ν ) n of solutions to problem (5.5) emanating from (λ ν 0 /f 0 , 0) and joining to (λ ν 0 /n, +∞).
Proof. By an argument similar to that of Theorem 5.4 and the conclusions of Theorem 5.6, we can prove the conclusion. Proof. Define By the conclusions of Theorem 5.2 and an argument similar to that of Theorem 5.6, we can obtain an unbounded component C σ ν of solutions to problem (1.4) such that (+∞, 0) ∈ C σ ν and (+∞, +∞) ∈ C σ ν . Finally, we show that there exists µ ν * > 0 such that problem (1.4) has no one-sign solutions for any λ ∈ (0, µ ν * ). Suppose on the contrary that there exists a sequence {λ n , u n } ∈ C σ ν such that lim Let v n = u n / u n . Obviously, one has v n = G p λ n m(x)f (u n (x)) u n p−1 .
By the compactness of F λ , we obtain that for some convenient subsequence v n → v 1 as n → +∞. Letting n → +∞, we obtain that v 0 ≡ 0. This contradicts v 1 = 1.
From the proof of Theorem 5.8, we can deduce the following corollary.

Uniqueness of positive solutions
In this section, under some more strict assumptions of f , we shall show that the unbounded continua which are obtained in Section 5 may be curves. We just show the case of f 0 = +∞ and f ∞ = 0. Other cases can be discussed similarly.
We get a contradiction.
Thus, without loss of generality, we can assume that there exists y 0 ∈ (0, x 0 ) such that u (y 0 ) > v (y 0 ). It is easy to see that there is an interval (α, β) such that u > v in (α, β) and u(x) = v(x) at x = α, β. Using an argument similar to that of Case 1, we have . This is a contradiction. Therefore, C + is a curve. Finally, we prove that u λ (x) is continuous with respect to λ. If λ 0 = 0, we define u 0 (x) ≡ 0. It is obvious that lim λ→0 u λ = 0. Next, we assume that λ 0 > 0. Let λ > 0 such that λ → λ 0 and u λ be the corresponding solutions. Then we have that u λ is bounded since C + does not blow up at a finite point. By the compactness of F λ , we obtain that for some convenient subsequence u λ → u in X. Clearly, we have u = u λ 0 . This completes the proof of the theorem. From Theorem 6.1, we can easily obtain the following result. Remark 6.1. In [27], the authors obtained the results similar to Theorem 6.1 under the assumptions of (A3), (A4) and (A6) f : [0, +∞) → (0, +∞) is nondecreasing, and there exists θ ∈ (0, 1) such that f (ks) ≥ k θ f (s) for k ∈ (0, 1) and u ∈ [0, +∞). (6.1) Obviously, we do not need that f to be nondecreasing in Theorem 6.1. In addition, (6.1) implies the assumption of (A5) with p = 2. To see this, letting 0 < s 1 < s 2 , we show that f (s 1 ) /s 1 > f (s 2 ) /s 2 . It is obvious that there exists a constant k ∈ (0, 1) such that s 1 = ks 2 .
Remark 6.2. In [27], the authors also proved that u λ is monotonic with respect to λ in the case of p = 2 and q(x) = ρ 2 . We conjecture that the solution u λ coming from Theorem 6.1 is also monotonic with respect to λ.

One-sign solutions without signum condition
In Section 5, we have studied the existence of one-sign solutions for (1.4) under the signum condition. Naturally, one may ask what will happen if f does not satisfy signum condition. In this section, we study problem (1.4) again but without signum condition.

Unilateral global bifurcation from infinity
In this subsection, we study unilateral global bifurcation phenomena from infinity for problem (1.1). Instead of (1.2), we assume that g satisfies lim |s|→+∞ g(x, s, λ) |s| p−1 = 0 (7.1) uniformly on [0, T ] and λ on bounded sets. We use S to denote the closure of the nontrivial solutions set of problem (1.1) in R × X. We add the points {(λ, ∞) λ ∈ R} to space R × X. Let S p denote the spectral set of problem (1.3).
The main result of this subsection is the theorem below.
Clearly, (7.2) is equivalent to It is obvious that (λ, 0) is always the solution of (7.3). By simple computation, we can show that the assumptions (7.

Global behavior of the components of one-sign solutions
In this subsection, we study the problem (1.4) again but without signum condition. We only consider the case of f 0 , f ∞ ∈ (0, +∞). Other cases can be discussed similarly. The details are left to the reader.
We shall obtain the results similar to ones of [39] for problem (1.4) in which the authors only studied the existence of positive solutions with p = 2. Note that in the case of p = 2, the authors of [39] also required that f ∈ C 2 (R, R) and satisfies f ′′ (s) < 0 for s ∈ [0, s 1 ). In this article, we drop these conditions completely. Hence, our results extend and improve the corresponding results of [39].
Proof. We only prove for the case (λ, u) ∈ C + + ∪ D + + since the other cases can be proved similarly. Suppose on the contrary that there exists (λ, u) ∈ C + + ∪ D + + such that either max{u(x) x ∈ [0, T ]} = s 1 or min{u(x) x ∈ [0, T ]} = s 2 . We only treat the case of max{u(x) x ∈ [0, T ]} = s 1 because the proof for the case of min{u(x) x ∈ [0, T ]} = s 2 can be given similarly.
Then, by an argument similar to that of Theorem 6.1, we can show that C σ + is a curve. Moreover, if (λ, u λ ) ∈ C ν + then u λ is continuous in λ. We conjecture that this result is also valid if the assumption (A4) is removed.