Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros

In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.


Introduction
Consider the following problem −∆ p u = λf (x, u) in Ω, u = 0 on ∂Ω, (1.1) where p ∈ (1, +∞), ∆ p u = div (|∇u| p−2 ∇u) is the p-Laplacian, λ > 0 is a real parameter, Ω is a bounded domain of R N with smooth boundary ∂Ω, and where f is a nonlinearity with two zeros which may vary in the x variable. We will investigate the existence and multiplicity of one-sign solutions of problem (1.1) involving a linear/superlinear growth nonlinearity by using of bifurcation and topological methods. Problems with linear/superlinear nonlinearities at infinity have been extensively studied. For the Laplacian, see for example [3,5,10,12,21,25,26,28,29,30,31,32,35,37,44,45,55]. For the p-Laplacian, see for example [6,14,46]. In most of these works, the nonlinearity is superlinear growth at infinity and satisfying the signum condition f (x, s)s > 0 for s = 0. The problem whose nonlinearity has nontrivial zeros has attracted a great deal of attention in history. In [37], this type of problems is considered for the Laplacian operator and a nonlinearity f that is independent on x, satisfying f (0) ≥ 0, f (β) = 0, and which is positive and superlinear for s > β > 0. Using topological degree arguments and under additional technical conditions which ensure a priori bounds, it is shown that there exist two positive solutions of problem (1.1). It is further shown that one solution lies strictly below β, while the other has a maximum greater than β. This type of problems was also studied in [40], where again the existence of two positive solutions of problem (1.1) was shown which improves the corresponding results of [37] in some sense. The characteristics of the problem are quite different when the nonlinearity is linear growth at infinity and has nontrivial zeros. In [5], by using of Rabinowitz's bifurcation Theorem, Ambrosetti and Hess studied the global behavior of the component of positive solutions of linear elliptic equation involving an asymptotically nonlinearity with positive zero. In [6], Ambrosetti et al. studied the existence of branch of positive solutions for asymptotically equidiffusive problem, which extends the corresponding ones of [5]. Delgado and Suárez [19] studied the existence of positive solutions for some indefinite nonlinear eigenvalue problems involving nonlinearities with positive zero. In [43], Ma studied the global behavior of the components of nodal solutions of one-dimensional asymptotically linear eigenvalue problems involving nonlinearities with two nontrivial zeros. Dai et al. [15] studied nodal solutions of quasilinear problems without signum condition. In the above papers, the zeros of nonlinearity are constants. In [27], mainly by variational method, the authors obtained some results involving the existence, multiplicity, and the behavior with respect to λ of positive solutions of problem (1.1) under some suitable assumptions on f . Motivated by the above papers, we study the existence and multiplicity of one-sign solutions of problem (1.1) by bifurcation and topological methods. However, the bifurcation results of [16,17,41,48,49,50] cannot be applied directly to quasilinear problems, which are very important tools in dealing with semilinear boundary value problems. So we first study the bifurcation phenomenon from infinity for the following abstract operator equation u = L(λ)u + K(λ, u), u ∈ E, (1.2) where λ varies in R, E is a real Banach space with norm · , the map λ → L(λ) is continuous. Moreover, we assume that L(·) : E → E is a homogeneous completely continuous operator and K : R × E → E is continuous with K(λ, u) = o ( u ) near u = ∞ uniformly on bounded λ intervals and u 2 K (λ, u/ u 2 ) is compact. If L(λ) = λL and L is linear, Rabinowitz [49] has shown that all characteristic values of L which are of odd multiplicity are bifurcation points from infinity. More precise, if µ is of odd multiplicity and S denotes the set of solutions of equation (1.2), then S possesses an unbounded component which meets (µ, ∞) and satisfies some alternatives. His method is to reduce the problem of bifurcation at infinity to bifurcation from the trivial solution axis which can be founded in [34,48]. We also refer to [7,11,33,38,39,41,42,51,52] and their references for the bifurcation from the trivial solution. Amann [2] established some bifurcation results from infinity in ordered Banach space by using this method.
As usually, µ is called an eigenvalue value of if there exists 0 ≡ ϕ ∈ E such that ϕ = L(µ)ϕ. Let Σ denote the set of real eigenvalues of equation (1.3). If µ ∈ Σ has geometric multiplicity 1, we let E 0 be a closed subspace of E such that where ϕ µ is an eigenfunction corresponding to µ with ϕ µ = 1. Let B r (0) = {u ∈ E : u < r}.
The first main result of this paper is the following theorem.
Theorem 1.1. If µ ∈ Σ is isolated, has geometric multiplicity 1 and such that for any ε > 0 small enough, then S possesses two unbounded components D + µ and D − µ which meet (µ, ∞). Moreover if Λ ⊂ R is an interval such that Λ ∩ Σ = {µ} and M is a neighborhood of (µ, ∞) whose projection on R lies in Λ and whose projection on E is bounded away from 0, then at least one of the following three properties is satisfied by D ν µ for ν = + and −: In order to deal with the case of superlinear growth at infinity, we establish a Whyburn type limit theorem as follows.
Theorem 1.2. Let X be a normal space and let {C n } be a sequence of unbounded connected subsets of X. Assume that: (i) there exists z * ∈ lim inf n→+∞ C n with z * = +∞; (ii) there exists a homeomorphism T : X → X such that T (z * ) < +∞ and {T (C n )} be a sequence of unbounded connected subsets of X; (iii) for every R > 0, ∪ +∞ n=1 T (C n ) ∩ B R is a relatively compact set of X.
Then D := lim sup n→+∞ C n is unbounded closed connected. Now, we are in the position to state the following five hypotheses on the nonlinearity f .
Remark 1.1. We do not require f satisfying the monotonicity hypothesis (M 1 ) of [27] which is essential to use sub and supersolution techniques. Moreover, we also do not require a > 0 being weakly p-superharmonic function. Due to these reasons, the conclusions of Theorem 1.3 improves the corresponding ones of [27, Theorem 1.1] in some sense even when f is nonnegative. Furthermore, we can get the exact multiplicity of one-sign solutions for problem (1.1) under more strict hypotheses than those of Theorem 1.3. Theorem 1.4. Let (H1)-(H3) and the following hypothesis hold, and such that f (·, s)/ϕ p (s) is decreasing in [0, a(·)] and is increasing in [b(·), 0].
Let λ 1,β denote the first eigenvalue of problem (1.5) with the weighted function β. By using of Theorem 1.1, the results involving the linear growth at infinity can be obtained as follows.
Theorem 1.5. Let (H1)-(H4) hold. Assume that λ 1,α > λ 1,β . Then (i) if λ ∈ (λ 1,β , λ 1,α ], then problem (1.1) has at least two solutions u + ∞ and u − ∞ , such that u + ∞ is positive in Ω and u − ∞ is negative in Ω; (ii) if λ ∈ (λ 1,α , +∞), then problem (1.1) has at least four solutions u + ∞ , u − ∞ , u + 0 and u − 0 such that u + ∞ and u + 0 (≤ a) are positive in Ω; u − ∞ and u − 0 (≥ b) are negative in Ω.  (ii) if λ ∈ (λ 1,β , +∞), then problem (1.1) has at least four solutions u + Our result involving the superlinear growth at infinity is the following theorem.  (ii) if λ ∈ (λ 1,α , +∞), then problem (1.1) has at least four solutions u +  [27]. The conditions (a)-(d) were used to prove that u + 0 < a in Ω which is fundamental to be able to obtain u + ∞ . The hypothesis (H 5 ) or (H 6 ) is required to prove that u + ∞ cannot stay below a. And (M 2 ) is necessary in [27] to use sub and supersolution techniques. We do not need these conditions or hypotheses because we use bifurcation method, and do not need u + 0 < a and the fact of u + ∞ cannot stay below a. We only care u + 0 ≡ u + ∞ . Of course, it can be guaranteed by u + 0 < a and the fact of u + ∞ cannot stay below a but these are not necessary.
Remark 1.6. The asymptotic behavior of solutions when λ → 0 or λ → λ 1,α is the natural result of bifurcation method. We also would like to point out that the solutions obtained in this paper are the sense of C 1 weak solutions. By standard elliptic regularity theory (see [22,24,54]), we know that any weak solution of problem (1.1) belongs to C 1,δ (Ω) with some constant δ ∈ (0, 1) under the hypotheses of (H1) and (H4).
Remark 1.7. We require a, b ∈ C 1 Ω because we need to use R × {a} and R × {b} to separate the bifurcation branches from the trivial solution axis and infinity. The reason to choose C 1 0 Ω as our state space rather than W 1,p 0 is that we need to compare the norm · ∞ and the norm of state space, which is easy to realize in C 1 0 Ω .
The rest of this paper is arranged as follows. In Section 2, we show or present some necessary preliminary results on bifurcation and topology, and give the proofs of Theorem 1.1 and Theorem 1.2. Section 3 is devoted to prove Theorem 1.3 and 1.4. In Section 4, we study the bifurcation phenomenon from infinity of problem (1.1) with linear growth nonlinearity at infinity and give the proofs of Theorem 1.5 and 1.6. In the last Section, we study the bifurcation phenomenon from infinity of problem (1.1) with superlinear growth nonlinearity at infinity and give the proofs of Theorem 1.7 and 1.8.

Preliminaries on bifurcation and topology
In this section, we first study the bifurcation phenomenon from infinity for equation (1.2) and prove Theorem 1.1. Then we give the proof of Theorem 1.2 and establish several topological results involving superior limit, which we believe could be useful and interesting by themselves.

Global bifurcation results
Suppose that H : R × E → E is completely continuous and H(λ, u) = o( u ) at u = 0 uniformly on bounded λ intervals. Let S be the closure of the set of nontrivial solution pairs of the following equation Firstly, we can get the following Rabinowitz type global bifurcation results.
Lemma 2.1. If µ ∈ Σ is isolated and such that (1.4) holds, then (µ, 0) is a bifurcation point for equation (2.1). Moreover, S possesses a maximal continuum C µ such that (µ, 0) ∈ C µ and C µ either Proof. Let us suppose by contradiction that (µ, 0) is not a bifurcation point. Then there exist ε > 0, ρ 0 > 0 such that for |λ − µ| ≤ 2ε and 0 < ρ < ρ 0 there is no nontrivial solution of the equation From the invariance of the degree under a compact homotopy we obtain that By taking ε smaller if necessary, we can assume that there is no real eigenvalue value of (1.3) in (µ, µ + 2ε]. Take λ = µ + ε. We claim that the equation has no solution u with u = ρ for every s ∈ [0, 1] and ρ sufficiently small. Suppose on the contrary, let {u n } be the solution of equation (2.3) with u n → 0 as n → +∞. Let v n = u n / u n . It follows that v n is a solution of the following problem By equation (2.4), we obtain that for some convenient subsequence v n → v 0 as n → +∞. Now v 0 verifies the following equation v = L(λ)v and v 0 = 1. This implies that λ = µ+ε is an eigenvalue value of (1.3). This is a contradiction. From the invariance of the degree under homotopies, we obtain that Similarly, we can find that Relations (2.2), (2.5) and (2.6) imply that This is a contradiction. Hence, (µ, 0) is a bifurcation point. By standard arguments in global bifurcation theory (see [48]), we can show the existence of a global branch of solutions of (2.1) emanating from (µ, 0).
From now on we assume that µ has geometric multiplicity 1. Let E = R × E. Given any ι ∈ R and 0 < s < +∞, we consider an open neighborhood of (ι, 0) in E defined by According to the Hahn-Banach theorem, there exists a linear functional l ∈ E * , here E * denotes the dual space of E, such that l (ϕ µ ) = 1 and E 0 = {u ∈ E : l(u) = 0}.
For any 0 < η < 1, define Applying the similar method to prove [41, Lemma 6.4.1] with obvious changes, we may obtain the following result.
there are s ∈ R and unique y ∈ E 0 such that u = sϕ µ + y and |s| > η u .
Furthermore, for these solutions (λ, u), 4) holds, then C µ possesses a subcontinuum in each of the cones Proof. According to Lemmata 2.1 and 2.2, the result is true for at least one of the cones. Suppose it is not true, for example, for K − η . Then, no continuum The new mapping H(λ, u) satisfies the same continuity properties as H(λ, u) and is an odd function of u. According to Lemma 2.1, the equation Φ(λ, u) = 0 possesses a component C µ of nontrivial solutions emanating from (µ, 0). By Lemma 2.2, there exists a constant δ 0 > 0 such that for each ̺ ∈ (0, δ 0 ), Due to the homotopy invariance of the degree, there exists It follows that for each 0 < ̺ < ̺ 1 , This is a contradiction.
µ,ε containing (µ, 0), and C ν µ to be the closure of ∪ 0<ε≤δ C ν µ,ε . Obviously, C ν µ is connected. In view of Lemma 2.2, the definition of C ν µ is independent from the choice of η and Lemma 2.4. If the assumptions of Lemma 2.1 hold and µ has geometric multiplicity 1, C µ can be decomposed into two sub-continua C + µ and C − µ , such that each of them either satisfies the alternatives of Lemma 2.1 or contains a point Proof. Suppose to the contrary that C − µ does not satisfy any of the alternatives of the statement. By argument similar to that of Theorem 6.4.3 of [41], we can show that there exists η 0 ∈ (0, 1) such that for each η ∈ (0, η 0 ), Now, consider the equation where Φ is the same as the proof of Lemma 2.3. Lemma 2.1 shows that a continuum C µ bifurcates from (µ, 0). According to (2.7) and the fact of Φ is odd in u, we find that It follows that C − µ is unbounded. A symmetric argument shows that C + µ also satisfies the alternatives of the statement. Now, using Lemma 2.1 and the method of [49], we can get the following theorem of bifurcation from infinity.

Topological results involving superior limit
The Whyburn's limit theorem [56, Theorem 9.1] is an important tool in the study of differential equations theory, see for example, [4,8,9,13] and references cited therein. However, if the collection of the infinite sequence of sets is unbounded, the Whyburn's limit theorem cannot be used directly because the collection may not be relatively compact.
For the convenience of the reader, we recall the definitions of superior limit and inferior limit here. Let G be any infinite collection of point sets, not necessarily different. The set of all points x of our space G such that every neighborhood of x contains points of infinitely many sets of G is called the superior limit or limit superior of G and is written lim sup G. The set of all points y such that every neighborhood of y contains points of all but a finite number of the sets of G is called the inferior limit or limit inferior of G and is written lim inf G.
To prove Theorem 1.2, we need the following topological lemma.
Lemma 2.5. Let X be a normal space and let {C n } be a sequence of unbounded connected subsets of X . Assume that: (i) there exists z * ∈ lim inf n→+∞ C n with z * < +∞; (ii) for every R > 0, ∪ +∞ n=1 C n ∩ B R is a relatively compact set of X Then D := lim sup n→+∞ C n is unbounded closed connected.
Proof. Let X R = X ∩ B R for any R > 0. Then X R is a metric subspace under the induced topology of X . Let A n = C n ∩ B R . Clearly, we have ∪ +∞ n=1 A n = ∪ +∞ n=1 C n ∩ B R . So ∪ +∞ n=1 A n is relatively compact in X R . Furthermore, z * ∈ lim inf n→+∞ C n implies that every neighborhood U (z * ) of z * contains points of all but a finite number of the sets of {C n }. So there exists a positive integer N such that for n > N, U (z * ) ∩ C n = ∅. Since z * < +∞, we can take R > 0 large enough such that U (z * ) ⊆ B R . Thus, U (z * ) ∩ A n = U (z * ) ∩ C n = ∅ for n > N. So we have z * ∈ lim inf n→+∞ A n . By Theorem 9.1 of [56], it follows that A = lim sup n→+∞ A n is connected in X R .
We claim that B := lim sup n→+∞ C n ∩ B R = A. For x ∈ A, then any neighborhood V in X R of x contains contains points of infinitely many sets of {C n ∩ B R }. So there exist x n i ∈ C n i ∩ B R such that x n i → x as i → +∞. It follows that x ∈ B R and x ∈ lim sup n→+∞ C n , i.e., x ∈ B. Conversely, if x ∈ B, any neighborhood V in X R of x contains a point z of lim sup n→+∞ C n and thus V , a neighborhood of z, contains points of infinitely many of the sets of C n ∩ B R . It follows that x ∈ A. Hence, B is connected. By the arbitrary of R, we get that D is connected. From [56], we know that D is closed.
Next, we show that D is unbounded. Suppose on the contrary that D is bounded. It is easy to see that D is a compact set of X by (ii) and the fact of z * ∈ D. Let U δ be a δ-neighborhood of D. So we have that (2.9) By (i) and the connectedness of C n , there exists an integer N 0 > 0, such that for all n > N 0 , C n ∩ ∂U δ = ∅. Take y n ∈ C n ∩ ∂U δ , then {y n : n > N 0 } is a relatively compact subset of X , so there exist y * ∈ ∂U δ and a subsequence {y n k } such that y n k → y * . The definition of superior limit shows that y * ∈ D. Therefore, y * ∈ ∂U δ ∩ D. However, this contradicts (2.9). Therefore, D is unbounded.
Proof of Theorem 1.2. We first claim that T (z * ) ∈ lim inf n→+∞ T (C n ). By (i), we have that T (z * ) ∈ T (lim inf n→+∞ C n ). It is enough to show that For any y ∈ lim inf n→+∞ T (C n ), let x = T −1 (y). For any neighborhood V in X of x, T (V ) is a neighborhood of y = T (x). So T (V ) contains points of all but a finite number of the sets {T (C n )}. Thus there exists m ∈ N such that y n ∈ T (V ) for any y n ∈ T (C n ) and n > m. Let x n = T −1 (y n ) for any n > m. Then obviously one has x n ∈ (C n ∩ V ) for any n > m. It follows that x ∈ lim inf n→+∞ C n . Therefore, y ∈ T (lim inf n→+∞ C n ). Conversely, for any y ∈ T (lim inf n→+∞ C n ), let x ∈ lim inf n→+∞ C n such that y = T (x). For any neighborhood V in X of y, T −1 (V ) is a neighborhood of x = T −1 (y). So T −1 (V ) contains points of all but a finite number of the sets {C n }. Thus there exists m ∈ N such that x n ∈ T −1 (V ) for any x n ∈ C n and n > m. Let y n = T (x n ) for any n > m. Then obviously one has y n ∈ (T (C n ) ∩ V ) for any n > m. It follows that y ∈ lim inf n→+∞ T (C n ). Now Lemma 2.5 shows that D = lim sup n→+∞ T (C n ) is unbounded closed connected. Let D = lim sup n→+∞ C n . Clearly, z * ∈ D. The unboundedness of z * shows that D is unbounded. From [56], we know that D is closed. So we only need to show that D is connected.
We next show that T (D) = D. For y ∈ D, then any neighborhood V in X of y contains points of infinitely many sets of {D n }. So there exist y n i ∈ D n i such that y n i → y as i → +∞. It follows that y ∈ lim sup n→+∞ D n . Let x n i ∈ C n i such that y n i = T (x n i ) and x = T −1 y. Then one has that x n i = T −1 (y n i ) → T −1 y = x. So x ∈ D, i.e., y ∈ T (D). Conversely, for any y ∈ T (D), let x ∈ D such that y = T (x). Then any neighborhood V in X of x contains contains points of infinitely many sets of {C n }. So there exist x n i ∈ C n i such that x n i → x as i → +∞. Let y n i ∈ D n i such that y n i = T (x n i ). It follows that y n i → y. So we have y ∈ D.
Finally, we show the connectedness of D. Suppose on the contrary that D is not bounded connected. So there exist two subsets A and B of D such that That is to say T (D) is not connected which contradicts the connectedness of D.
Note that the unboundedness of {T (C n )} is not necessary in the conclusion of Theorem 1.2. In fact, by Theorem 9.1 of [56], in view of the proof of Theorem 1.2, we can immediately get the following corollary.
Corollary 2.1. Let X be a normal space and let {C n } be a sequence of unbounded connected subsets of X. Assume that: (i) there exists z * ∈ lim inf n→+∞ C n with z * = +∞; (ii) there exists a homeomorphism T : X → X such that T (z * ) < +∞; (iii) ∪ +∞ n=1 T (C n ) is a relatively compact set of X.
Then D := lim sup n→+∞ C n is unbounded closed connected.
The following lemma describes a relation of its superior limit and elements of a sequence of sets, which is very useful to understand superior limit via element because superior limit may be more obscure than concrete element. Lemma 2.6 (see [56]). Let (X, ρ) be a metric space. If {A i } i∈N is a sequence of sets of X whose limit superior is L and the sum of whose elements is a conditionally compact set, then for each ǫ > 0 there exists an m such that for every n > m, A n ⊂ V ǫ (L), where V ǫ (L) denotes the set of all points p with ρ(p, x) < ǫ for any x ∈ L.
In concrete problems, the collection of the infinite sequence of sets may not be relatively (conditionally) compact. So the following two propositions are needed, which will play a key role in the analysis of the structure of bifurcation branch from infinity (see Section 5).
Proposition 2.2. If {C i } i∈N is a sequence of sets whose limit superior is L and for every R > 0, ∪ +∞ i=1 C i ∩ B R is a relatively compact set, then for each ǫ > 0 there exists an m such that for every n > m, C n ⊂ V ǫ (L).
Proof. Let X R = X ∩ B R for any R > 0. Then X R is a metric subspace under the induced topology of X.
For any x ∈ A, then any neighborhood V in X R of x contains points of infinitely many sets of {C n ∩ B R }. So there exist x n i ∈ C n i ∩ B R such that x n i → x as i → +∞. It follows that x ∈ B R and x ∈ lim sup n→+∞ C n , i.e., x ∈ B. Conversely, if x ∈ B, any neighborhood V in X R of x contains a point z of lim sup n→+∞ C n and thus V , a neighborhood of z, contains points of infinitely many of the sets of C n ∩ B R . It follows that x ∈ A. Now by Lemma 2.6, for each ǫ > 0 there exists an m such that for every n > m, A n ⊂ V ǫ (B) in X R . Clearly, one has V ǫ (B) ⊂ V ǫ (L). Thus we have A n ⊂ V ǫ (L) in X. By the arbitrary of R, we get that C n ⊂ V ǫ (L) in X for every n > m. Proposition 2.3. If {C i } i∈N is a sequence of sets whose limit superior is L and there exists a homeomorphism T : X → X such that for every R > 0, ∪ +∞ i=1 T (C i ) ∩ B R is a relatively compact set, then for each ǫ > 0 there exists an m such that for every n > m, C n ⊂ V ǫ (L).
Proof. By an argument similar to that of Theorem 1.2, we can obtain that lim sup n→+∞ T (C n ) = T (L). Applying Proposition 2.2 to {T (C i )} i∈N , we get that for each ǫ > 0 there exists an m such that for every n > m, T (C n ) ⊂ V ǫ (T (L)). For any fixed x 0 ∈ C n , we have T (x 0 ) ∈ T (C n ). Clearly, one has T (x 0 ) ∈ V ǫ (T (L)). It means that for any z ∈ L, T (x 0 ) − T (z) < ǫ. It follows from the continuity of T −1 that x 0 − z = T −1 (T (x 0 )) − T −1 (T (z)) < ǫ for each ǫ > 0. By the arbitrary of x 0 , we get that C n ⊂ V ǫ (L) for every n > m.

Bifurcation from trivial solution axis
The aim of this section is to prove Theorem 1.3 and 1.4. Firstly, we define and consider the following problem Let E = u ∈ C 1 (Ω) : u = 0 on ∂Ω with the usual norm u = max Ω |u| + max Ω |∇u|. Consider the following auxiliary problem for a given h ∈ L r/(r−1) (Ω), where r ∈ (1, p * ). By a solution of problem (3.2) we understand that u ∈ W 1,p 0 (Ω) satisfying problem (3.2) in the weak sense, i.e., such that Ω |∇u| p−2 ∇u∇v dx = Ω hv dx for all v ∈ W 1,p 0 (Ω). We have known that for every given h ∈ L r/(r−1) (Ω) there is a unique solution u to problem (3.2) (see [20]). Let R p (h) denote the unique solution to problem (3.2) for a given h ∈ L r/(r−1) (Ω). It is well known that R p : L ∞ (Ω) → E is completely continuous (see [22,23,36]). The definition of f implies that u ∈ C 1,δ Ω with some constant δ ∈ (0, 1) for every weak solution u of problem (3.1).
Next we study the bifurcation phenomenon of problem (3.1) from the trivial solution axis.
Thus, u * ∈ C 1,δ Ω and does not change sign in Ω, say u * ≥ 0 in Ω. By the strong maximum principle of [47], we have u * > 0 in Ω. Obviously, u * cannot be approximated by any sequence of S − . This contradicts our assumption. Arguing in the same way leads to C − ⊆ ({(λ 1,α , 0)} ∪ (R × S − )). Let us show that both C + and C − are unbounded. Consider the following auxiliary problem where ξ is defined by The previous argument shows that an unbounded continuum C bifurcates from (λ 1,α , 0) and can be split into C + and C − with C ν connected, C ν ⊆ ({(λ 1,α , 0)} ∪ (R × S ν )). It is easy to see that C − = − C + . It follows that both C + and C − are unbounded. It is clear that C + ⊆ C + . Therefore, C + must be unbounded. A symmetric argument shows that C − is also unbounded.
It is easy to verify that (λ 1,α , 0) is the unique bifurcation point of one-sign solutions of problem (3.1) from the trivial solution axis. Now, we can prove Theorem 1.3 by virtue of Proposition 3.1.
Proof of Theorem 1.3. We claim that u is also a solution of problem (1.1) for any (λ, u) ∈ C which is obtained in Proposition 3.1. It is enough to show that u ∈ [b(x), a(x)] for any (λ, u) ∈ C . In fact if there exists x 0 ∈ Ω such that u (x 0 ) > a (x 0 ) or u (x 0 ) < b (x 0 ), the definition of f will imply that u ≡ 0. This is impossible. So one has u ∈ [b(x), a(x)] for any (λ, u) ∈ C . It follows that u is also a solution of problem (1.1) for any (λ, u) ∈ C .
Next we show that the projection of C + on R is unbounded. It is sufficient to show that the set {(λ, u) ∈ C + : λ ∈ (0, d]} is bounded for any fixed d ∈ (0, +∞). Suppose on the contrary that there exists (λ n , u n ) ∈ C + , n ∈ N, such that λ n → µ ≤ d, u n → +∞ as n → +∞. Let w n = u n / u n . Then we have that Clearly, we have that where M = max x∈Ω a(x). It means that as n → +∞. By the compactness of R p , we obtain that for some convenient subsequence w n → w 0 as n → +∞. Letting n → +∞, we obtain that w 0 ≡ 0. This contradicts w 0 = 1. This together with the fact that C + joins (λ 1,α , 0) to infinity yields that where Proj (C + ) denotes the projection of C + on R. Similarly, we can show that Now the existence of u + 0 and u − 0 can be got from Proposition 3.1 and the above argument immediately. Moreover, we have u + 0 ≡ a and u − 0 ≡ b on Ω, that is to say, there exist some x 0 , x 1 ∈ Ω such that u + 0 (x 0 ) < a (x 0 ) and u − 0 (x 1 ) > b (x 1 ). Otherwise, it contradicts hypothesis (H2).
To prove Theorem 1.4, we propose the definition of linearly stable solution. For any φ ∈ E and nontrivial solution u of problem (1.1), the linearized problem of problem (1.1) about u at the direction φ is By some simple computations, we can show that (H6) follows that (p − 1)f (x, s) − f s (x, s)s ≥ 0 for any s ≥ 0. Since u > 0 and φ 1 > 0 in Ω, then µ 1 > 0 and the positive solution u must be stable.
To complete the proof, it suffices to show that u + (λ, ·) (u − (λ, ·)) is increasing (decreasing) with respect to λ. We only prove the case of u + (λ, ·). The proof of u − (λ, ·) can be given similarly. Since u + (λ, ·) is differentiable with respect to λ (as a consequence of Implicit Function Theorem), then du + (λ,·) By an argument similar to that of Lemma 3.1, we can show that

Linear growth at infinity
In this section, we consider the case of linear growth of f at infinity. We first use Theorem 1.1 to study the bifurcation phenomenon of problem (1.1) from infinity. Then we present the proofs of Theorem 1.5 and 1.6. Proof. Let η : Ω × R → R be such that Let us consider −∆ p u = λβ(x)ϕ p (u) + λη(x, u) in Ω, u = 0 on ∂Ω as a bifurcation problem from infinity. Obviously, problem (1.1) can be equivalently written as where K p (u) = βϕ p (u), H(λ, ·) denotes the usual Nemitsky operator associated with λη. Let L(λ)u = R p λ K p u and H(λ, u) = R p λ K p u + H(λ, u) − L(λ)u. Then it is easy to see that L(·) : E → E is homogeneous completely continuous. From hypotheses (H1) and (H4), the right hand side of equation ( |η(x, s)| for any x ∈ Ω, then η is nondecreasing with respect to u. Define |η(x, s)| for any x ∈ Ω.
Then we can see that lim u→+∞ η(x, u) u p−1 = 0 and η(x, u) ≤ η x, It follows that Further it follows from (4.2) that uniformly in x ∈ Ω. It follows that H = o( u ) near u = ∞ uniformly on bounded λ intervals. By an argument similar to that of [20, Proposition 2.2], we can get that where λ 2,β denotes the second eigenvalue of problem (1.5) with the weighted function β.
Applying Proposition 2.1 to problem (4.1), there exists a continuum D of solutions of problem (1.1) meeting (λ 1,β , ∞) and satisfying at least one of the alternatives of Proposition 2.1. We claim that 1 o of Proposition 2.1 does not occur. Otherwise, D will crosse the line R × {a} or R × {b} in R × E. That is to see a or b is a solution of problem (1.1) which contradicts the assumption of (H2). So 2 o of Proposition 2.1 occurs.
It is easy to verify that (λ 1,β , ∞) is the unique bifurcation point of one-sign solutions of problem (1.1) from infinity. Now, we can prove Theorem 1.5 and 1.6 by using of Proposition 4.1.
Proof of Theorem 1.5. The conclusion (i) is a direct corollary of Proposition 4.1. So we only need to show the conclusion (ii). Now, noting Proposition 3.1 and 4.1, it is sufficient to show that C ν ∩ D ν = ∅. Otherwise, D ν meets R = {(λ, 0) : λ ∈ R}. Hence, it will crosse the line R × {a} or R × {b} in R × E. That is to see a or b is a solution of problem (1.1) which contradicts hypothesis (H2).
Proof of Theorem 1.6. The proof is similar to that of Theorem 1.5. So we omit it here.

Superlinear growth at infinity
In this section, we consider the case of superlinear growth of f at infinity. We first use Theorem 1.2 and Proposition 4.1 to study the bifurcation phenomenon of problem (1.1) from infinity. Then we give the proof of Theorem 1.7. Finally, we study the asymptotic behavior of the solutions when λ → +∞ and prove Theorem 1.8. Proposition 4.1 implies that there exist two sequence unbounded continua D ν n of solution set of problem (5.1) emanating from (λ 1,β /n, ∞).
Taking z * = (0, ∞), clearly z * ∈ lim inf n→+∞ D ν n with z * R×E = +∞. Define mapping T : X → X such that It is easy to verify that T is homeomorphism and T (z * ) R×E = 0. The compactness of R p implies that ∪ +∞ n=1 T (D ν n ) ∩ B R is pre-compact. Theorem 1.2 implies that D ν = lim sup n→+∞ D ν n is unbounded closed connected such that z * ∈ D ν . Next we show that the projection of D ν on R is unbounded. From the argument of Proposition 4.1, we have known that D ν n has unbounded projections on R. By Proposition 2.3, for each ǫ > 0 there exists an m such that for every n > m, D ν n ⊂ V ǫ (D ν ). This implies that λ 1,β n , +∞ ⊆ Proj (D ν n ) ⊆ Proj (V ǫ (D ν )) .
It follows that the projection of D ν is unbounded on R. Therefore, we have that (0, +∞) = Proj (D ν ) .
Proof of Theorem 1.7. In view of Proposition 5.1, an argument similar to that of Theorem 1.5 can yield the desired conclusions.
In order to prove Theorem 1.8, we need the following priori estimate for the solution of problem (1.1).
Proof. The conclusion involving the case of u being positive is just Lemma 4.1 of [27]. Moreover, the negative case is similar to the positive case.
Furthermore, we also need the following Liouville-type theorem for a non-positive function with zero. (h2) There exist constants γ > 0 and q ∈ (p, p * ) such that h(s) ≤ −γ b − s q−1 for s < b.
Then any C 1 weak solution of the problem is either the constant function w ≡ 0, or else w ≡ b.
Proof. Let w = −v and f (t) = −h(−t). Applying Theorem 1.8 of [27] to −∆ p w = f (w) in R N , w ≥ 0, the desired conclusion can be got immediately.