Abstract
We consider the boundary value problem
where:
(1) m ≥ 3, η_{i }∈ (0, 1) and α_{i }> 0 with ;
(2) g : ℝ → ℝ is continuous and satisfies
and
(3) p : [0, 1] × ℝ^{2 }→ ℝ is continuous and satisfies
for some C > 0 and β ∈ (0, 1/2).
We obtain infinitely many solutions having specified nodal properties by the bifurcation techniques.
MSC(2000). 34B15, 58E05, 47J10
Keywords:
Nodal solutions; Second order equations; Multipoint boundary value problems; Bifurcation1 Introduction
We consider the nonlinear boundary value problem
where
(H1) m ≥ 3, η_{i }∈ (0, 1) and α_{i }> 0 with
(H2) g : ℝ → ℝ is continuous and satisfies
and
(H3) p : [0, 1] × ℝ^{2 }→ ℝ is continuous and satisfies
for some C > 0 and β ∈ (0, 1/2).
In order to state our results, we first recall some standard notations to describe the nodal properties of solutions. For any integer, n ≥ 0, C^{n}[0, 1] will denote the usual Banach space of ntimes continuously differentiable functions on [0, 1], with the usual suptype norm, denoted by  · n. Let X := {u ∈ C^{2}[0, 1]: u satisfies (1.2)}, Y := C^{0}[0, 1], with the norms  · _{2 }and  · _{0}, respectively. Let
with the norms  · _{E}.
We define a linear operator L : X → Y by
In addition, for any continuous function g : ℝ → ℝ and any u ∈ Y, let g(u) ∈ Y denote the function g(u(x)), x ∈ [0, 1].
Next, we state some notations to describe the nodal properties of solutions of (1.1), see [1] for the details. For any C^{1 }function u, if u(x_{0}) = 0, then x_{0 }is a simple zero of u, if u'(x_{0}) ≠ 0. Now, for any integer k ≥ 1 and any ν ∈ {+, }, we define sets consisting of the set of functions u ∈ C^{2}[0, 1] satisfying the following conditions:
(i) u(0) = 0, νu'(0) > 0; (ii) u has only simple zeros in [0, 1] and has exactly k  1 zeros in (0, 1).
(i) u(0) = 0, νu'(0) > 0; (ii) u' has only simple zeros in (0, 1) and has exactly k such zeros; (iii) u has a zero strictly between each two consecutive zeros of u'.
Remark 1.1 If we add the restriction u' (1) ≠ 0 on the functions in then becomes the set , which used in [1]. The reason we use rather than is that the Equation (1.1) is not autonomous anymore.
In [1, Remarks 2.1 and 2.2], Rynne pointed out that
a. If , then u has exactly one zero between each two consecutive zeros of u', and all zeros of u are simple. Thus, u has at least k  1 zeros in (0, 1), and at most k zeros in (0, 1];
b. The sets are open in X and disjoint;
c. When considering the multipoint boundary condition (1.2), the sets are in fact more appropriate than the sets .
The main result of this paper is the following
Theorem 1.1 Let (H1)(H3) hold. Then there exists an integer k_{0 }≥ 1 such that for all integers k ≥ k_{0 }and each ν ∈ {+, } the problem (1.1), (1.2) has at least one solution .
Superlinear problems with classical boundary value conditions have been considered in many papers, particularly in the second and fourth order cases, with either periodic or separated boundary conditions, see for example [211] and the references therein. Specifically, the second order periodic problem is considered in [2,3], while [47] consider problems with separated boundary conditions, and results similar to Theorem 1.1 were obtained in each of these papers. The fourth order periodic problem is considered in [810]. Rynne [11] and De Coster [12] consider some general higher order problems with separated boundary conditions also.
Calvert and Gupta [13] studied the superlinear threepoint boundary value problem
(which is a nonlocal boundary value problem), under the assumptions:
(A0) β ∈ (0, 1) ∪ (1, ∞);
(A1) g : ℝ → ℝ is continuous and satisfies g(s)s > 0, s ≠ 0, is increasing and
(A2) p : [0, 1] × ℝ^{2 }→ ℝ is a function satisfying the Carathéodory conditions and satisfies
where M_{1 }: [0, 1] × [0, ∞) → [0, ∞) satisfies the condition: for each s ∈ [0, ∞), M_{1}(·, s) is integrable on [0, 1] and for each t ∈ [0, 1], M_{1}(t, ·) is increasing on [0, ∞) with as s → ∞.
Calvert and Gupta used LeraySchauder degree and some ideas from Henrard [14] and Cappieto et al. [5] to prove the existence of infinity many solutions for (1.7), (1.8). Their results extend the main results in [14].
It is the purpose of this paper to use the global bifurcation theorem, see [15] and [1], to obtain infinity many nodal solutions to mpoint boundary value problems (1.1), (1.2) under the assumptions (H1)(H3). Obviously, our conditions (H2) and (H3) are much weaker than the corresponding restrictions imposed in [13]. Our paper uses some of ideas of Rynne [10], which deals with fourth order twopoint boundary value problems. By the way, the proof [10, Lemma 2.8] contains a small error (since u″_{0 }≥ ζ_{4}(0) ⇏ u″_{0 }≥ ζ_{4}(R) there). So, we introduce a new function χ (see (3.7)) with
which are required in applying Lemma 3.4.
2 Eigenvalues of the linear problem
First, we state some preliminary results related to the linear eigenvalue problem
Denote the spectrum of L by σ(L). The following spectrum results on (2.1) were established by Rynne [1], which extend the main result of Ma and O'Regan [16].
Lemma 2.1. [1, Theorem 3.1] The spectrum σ(L) consists of a strictly increasing sequence of eigenvalues λ_{k }> 0, k = 1, 2, ..., with corresponding eigenfunctions . In addition,
(i) lim_{k→∞ }λ_{k }= ∞;
(ii) , for each k ≥ 1, and ϕ_{1 }is strictly positive on (0, 1).
Lemma 2.2 [1, Theorem 3.8] For each k ≥ 1, the algebraic multiplicity of the characteristic value λ_{k }of L^{1 }: Y → Y is equal to 1.
3 Proof of the main results
For any u ∈ X, we define e(u)(·): [0, 1] → ℝ by
It follows from (1.5) that
For any s ∈ ℝ, let
and for any s ≥ 0, let
We now consider the boundary value problem
where α ∈ [0, 1] is an arbitrary fixed number and λ ∈ ℝ. In the following lemma (λ, u) ∈ ℝ × X will be an arbitrary solution of (3.2).
By (H2), we can choose b_{1 }≥ 1 such that
By (1.2), we have the following
Lemma 3.1. Let (H1) hold and let u ∈ X. Then
Lemma 3.2. Let u be a solution of (3.2). Then for any x_{0}, x_{1 }∈ [0, 1],
Proof. Multiply (3.2) by u' and integrate from x_{0 }to x_{1}, then we get the desired result. ■
In the following, let us fix R ∈ (0, ∞) so large that R ≥ b_{1 }and
Lemma 3.3. There exists an increasing function ζ_{1 }: [0, ∞) → [0, ∞), such that for any solution u of (3.2) with 0 ≤ λ ≤ R and u(x0) + u'(x_{0}) ≤ R for some x_{0 }∈ [0, 1], we have
Proof. Choose x_{1 }∈ [0, 1] such that u'_{0 }= u'(x_{1}). We obtain from Lemma 3.2 that
Combining this with (3.1), (3.4), it concludes that
with
This implies
■
Define
Clearly, the function is nondecreasing.
Lemma 3.4 Let u be a solution of (3.2) with 0 ≤ λ ≤ R and u'_{0 }≥ ζ_{2}(R) for some R > 0. Then, for any x ∈ [0, 1] with u(x) ≤ R, we have u'(x) ≥ R^{2}.
Proof. Suppose, on the contrary that there exists x_{0 }∈ (0, 1) such that u(x_{0}) ≤ R and u'(x_{0}) < R^{2}. Then
Combining this with λ ≤ R < R + R^{2 }and using Lemma 3.3, it concludes that
However, this is impossible if u'_{0 }≥ ζ_{2}(R). ■
For fixed R > b_{1}, let us define
Let us now consider the problem
where θ : ℝ → ℝ is a strictly increasing, C^{∞}function with θ(s) = 0, s ≤ 1 and θ(s) = 1, s ≥ 2. The nonlinear term in (3.8) is a continuous function of (λ, u) ∈ ℝ × X and is zero for λ ∈ ℝ, u'_{0 }≤ χ(λ), so (3.8) becomes a linear eigenvalue problem in this region, and overall the problem can be regarded as a bifurcation (from u = 0) problem.
The next lemma now follows immediately.
Lemma 3.5 The set of solutions (λ, u) of (3.8) with u'_{0 }≤ χ(λ) is
We also have the following global bifurcation result for (3.8).
Lemma 3.6 For each k ≥ 1 and ν ∈ {+, }, there exists a connected set of nontrivial solutions of (3.8) such that is closed and connected and:
(i) there exists a neighborhood N_{k }of (λ_{k}, 0) in ℝ × E such that ,
(ii) meets infinity in ℝ × E (that is, there exists a sequence , such that λ_{n} + u_{n}_{E }→ ∞).
Proof. Since L^{1 }: Y → X exists and is bounded, (3.8) can be rewritten in the form
and since L^{1 }can be regarded as a compact operator from Y to E, it is clear that finding a solution (λ, u) of (3.8) in ℝ × E is equivalent to finding a solution of (3.9) in ℝ × E. Now, by the similar method used in the proof of [1, Theorem 4.2]), we may deduce the desired result.
■
Since e(u)(t) σ 0 in (3.8), nodal properties need not be preserved. However, we will rely on preservation of nodal properties for "large" solutions, encapsulated in the following result.
Lemma 3.7 If (λ, u) is a solution of (3.8) with λ ≥ 0 and u'_{0 }> χ(λ), then , for some k ≥ 1 and ν ∈ {+, }.
Proof. If for any k ≥ 1 and ν, then one of the following cases must occur:
Case 1. u'(0) = 0;
Case 2. u' (τ) = u″(τ) = 0 for some τ ∈ (0, 1].
In the Case 1, u(t) ≡ 0 on [0, 1]. This contradicts the assumption u'_{0 }> χ(λ) ≥ ζ_{2}(λ). So this case cannot occur.
In the Case 2, we have from (3.8) that
Since u'_{0 }> χ(λ), we have from the definition of θ that
It follows from Lemma 3.4 that u(τ) > R ≥ b_{1}. Combining this with (3.11) and (3.3), it concludes that
which contradicts (3.10). So, Case 2 cannot occur.
Therefore, for any k ≥ 1 and ν ∈ {+, }. ■
In view of Lemmas 3.5 and 3.7, in the following lemma, we suppose that (λ, u) is an arbitrary nontrivial solution of (3.8) with λ ≥ 0 and , for some k ≥ 1 and ν.
Lemma 3.8. There exists an integer k_{0 }≥ 1 (depending only on χ(0)) such that for any nontrivial solution u of (3.8) with λ = 0 and χ(0) ≤ u'_{0 }≤ 2χ(0), we have
Proof. Let x_{1}, x_{2 }be consecutive zeros of u. Then there exists x_{3 }∈ (x_{1}, x_{2}) such that u'(x_{3}) = 0, and hence, Lemma 3.4, (3.3), and (3.7) yield that u(x_{3}) > 1. Since
for some τ_{1 }∈ (x_{3}, x_{2}), τ_{2 }∈ (x_{1}, x_{3}), it follows that
Notice that u'_{0 }> χ(0) ≥ ζ_{2}(R) implies that for some k ∈ ∞ and ν ∈ {+, }, and subsequently, there exist 0 < r_{1 }< r_{2 }< · · · < r_{k1}, such that
This together with (3.14) imply that
and accordingly, k < u'_{0}/2 + 1 ≤ χ(0) + 1. ■
Now let
Lemma 3.9. Suppose that 0 ≤ λ ≤ R and u'_{0 }≥ χ(R). Then W_{R}(u) consists of at least k intervals and at most k + 1 intervals, each of length less than 2/R, and V_{R}(u) consists of at least k intervals and at most k + 1 intervals.
Proof. Lemma 3.4 implies that u'(x) ≥ R^{2 }for all x ∈ W_{R}(u). For any interval I ⊂ W_{R}(u), u' does not change sign on I, say,
We claim that the length of I is less than 2/R.
In fact, for x, y ∈ I with x > y, say,
Thus,
which implies
The case
can be treated by the similar method. Since u is monotonic in any subinterval containing in W_{R}(u), the desired result is followed. ■
Lemma 3.10. There exists ζ_{3 }with lim_{R→∞ }ζ_{3}(R) = 0, and η_{1 }≥ 0 such that, for any R ≥ η_{1}, if either
(a) 0 ≤ λ ≤ R and u'_{0 }= 2χ(R), or
(b) λ = R and χ(R) ≤ u'_{0 }≤ 2χ(R),
then the length of each interval of V_{R}(u) is less than ζ_{3}(R).
Proof. Define H = H(R) by
and let ζ_{3}(R) := 2π/H(R). By (1.4), lim_{R→∞ }H(R) = ∞, so lim_{R→∞}ζ_{3}(R) = 0, and we may choose η_{1 }≥ b_{1 }sufficiently large that H(R) > 0 for all R ≥ η_{1}.
We firstly show that
In fact, if u(x) ≤ R on [0, 1], then Lemma 3.4 yields that either
or
However, these contradict the boundary conditions (1.2), since (H1) implies u'(s_{0}) = 0 for some s_{0 }∈ (0, 1). Therefore, (3.15) is valid.
Now, Let us choose x_{0}, x_{2 }such that either
(1) u(x_{0}) = u(x_{2}) = R and u > R on (x_{0}, x_{2}) or
(2) u(x_{0}) = R, x_{2 }= 1 and u > R on (x_{0}, 1].
(the case of intervals on which u < 0 is similar). Let
By (3.8) and the construction of H(R), if either (a) or (b) holds then
and by Lemma 3.4, u'(x_{0}) > 0, and u'(x_{2}) < 0, if x_{2 }< 1.
Suppose, now on the contrary that x_{2 } x_{0 }> ζ_{3}(R), that is, l := 2π/(x_{2 } x_{0}) < H(R). Defining x_{1 }= (x_{0 }+ x_{2})/2 and
we have
and hence
and this contradiction shows that x_{2 } x_{0 }≤ ζ_{3}(R), which proves the lemma.
■
Now, we are in the position to prove Theorem 1.1.
Proof of Theorem 1.1 Now, choose an arbitrary integer k ≥ k_{0 }and ν ∈ {+, }, and choose Λ > max{η_{1}, μ_{k}} (Here, we assume Λ > η_{1}, so that Lemma 3.10 could be applied!) such that
Notice that Lemma 3.9 implies that if u'_{0 }≥ χ(Λ), then the length of each interval of W_{Λ}(u) is less than for 0 ≤ λ ≤ Λ. This together with (3.16) and Lemma 3.10 imply that there exists no solution (λ, u) of (3.8), which satisfies either
(a) 0 ≤ λ ≤ Λ and u'_{0 }= 2χ(Λ) or
(b) λ = Λ and χ(Λ) ≤ u'_{0 }≤ 2χ(Λ).
Now, let us denote
It follows from Lemma 3.5 that "enters" B through the set D_{1}, while from Lemma 3.7, . Thus, by Lemma 3.6 and the fact
must "leave" B. (Suppose, on the contrary that does not "leave" B, then
which contradicts the fact that joins (μ_{k}, 0) to infinity in ℝ × E.) Since is connected, it must intersect ∂B. However, Lemmas 3.83.10 (together with (3.16)) show that the only portion of ∂B (other than D_{1}), which can intersect is D_{2}. Thus, there exists a point , and clearly provides the desired solution of (1.1)(1.2).
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to the anonymous referee for his/her valuable suggestions. Supported by the NSFC(No.11061030), the Fundamental Research Funds for the Gansu Universities.
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