Abstract
We shall study the existence and multiplicity of nodal solutions of the nonlinear secondorder twopoint boundary value problems,
The proof of our main results is based upon bifurcation techniques.
Mathematics Subject Classifications: 34B07; 34C10; 34C23.
Keywords:
nodal solutions; bifurcation1 Introduction
In [1], Ma and Thompson were considered with determining interval of μ, in which there exist nodal solutions for the boundary value problem (BVP)
under the assumptions:
(C1) w(·) ∈ C([0, 1], [0, ∞)) and does not vanish identically on any subinterval of [0, 1];
(C2) f ∈ C(ℝ, ℝ) with sf(s) > 0 for s ≠ 0;
(C3) there exist f_{0}, f∞ ∈ (0, ∞) such that
It is well known that under (C1) assumption, the eigenvalue problem
has a countable number of simple eigenvalues μ_{k}, k = 1, 2,..., which satisfy
and let μ_{k }be the kth eigenvalue of (1.2) and φ_{k }be an eigenfunction corresponding to μ_{k}, then φ_{k }has exactly k  1 simple zeros in (0,1) (see, e.g., [2]).
Using Rabinowitz bifurcation theorem, they established the following interesting results:
Theorem A (Ma and Thompson [[1], Theorem 1.1]). Let (C1)(C3) hold. Assume that for some k ∈ ℕ, either
In [3], Ma and Thompson studied the existence and multiplicity of nodal solutions for BVP
They gave conditions on the ratio
Using Rabinowitz bifurcation theorem also, they established the following two main results:
Theorem B (Ma and Thompson [[1], Theorem 2]). Let (C1)(C3) hold. Assume that either (i) or (ii) holds for some k ∈ ℕ and j ∈ {0} ∪ ℕ;
(i) f_{0 }< μ_{k }< ⋯ < μ_{k+j }< f_{∞};
(ii) f_{∞ }< μ_{k }< ⋯ < μ_{k+j }< f_{0},
where μ_{k }denotes the kth eigenvalue of (1.2). Then BVP (1.3) has 2(j + 1) solutions
Theorem C (Ma and Thompson [[1], Theorem 3]). Let (C1)(C3) hold. Assume that there exists an integer k ∈ ℕ such that
where μ_{k }denotes the kth eigenvalue of (1.2). Then BVP (1.3) has no nontrivial solution.
From above literature, we can see that the existence and multiplicity results are largely based on the assumption that t and u are separated in nonlinearity term. It is interesting to know what will happen if t and u are not separated in nonlinearity term? We shall give a confirm answer for this question.
In this article, we consider the existence and multiplicity of nodal solutions for the nonlinear BVP
under the following assumptions:
(H_{1})
(H_{2})
(H_{3}) f(t, s)s > 0 for t ∈ (0, 1) and s ≠ 0.
Remark 1.1. From (H_{1})(H_{3}), we can see that there exist a positive constant ϱ and a subinterval [α, β] of [0, 1] such that α < β and
In the celebrated study [4], Rabinowitz established Rabinowitz's global bifurcation theory [[4], Theorems 1.27 and 1.40]. However, as pointed out by Dancer [5,6] and LópezGómez [7], the proofs of these theorems contain gaps, the original statement of Theorem 1.40 of [4] is not correct, the original statement of Theorem 1.27 of [4] is stronger than what one can actually prove so far. Although there exist some gaps in the proofs of Rabinowitz's Theorems 1.27, 1.40, and 1.27 has been used several times in the literature to analyze the global behavior of the component of nodal solutions emanating from u = 0 in wide classes of boundary value problems for equations and systems [1,2,8,9]. Fortunately, LópezGómez gave a corrected version of unilateral bifurcation theorem in [7].
By applying the bifurcation theorem of LópezGómez [[7], Theorem 6.4.3], we shall establish the following:
Theorem 1.1. Suppose that f(t, u) satisfies (H_{1}), (H_{2}), and (H_{3}), then problem (1.4) possesses two solutions
Similarly, we also have the following:
Theorem 1.2. Suppose that f(t, u) satisfies (H_{3}) and
Remark 1.2. We would like to point out that the assumptions (H_{1}) and (H_{2}) are weaker than the corresponding conditions of Theorem A. In fact, if we let f(t, s) ≡ μw(t)f(s), then we can get
Using the similar proof with the proof Theorems 1.1 and 1.2, we can obtain the more general results as follows.
Theorem 1.3. Suppose that (H_{3}) holds, and either (i) or (ii) holds for some k ∈ ℕ and j ∈ {0} ∪ ℕ:
(i)
(ii)
Then BVP (1.4) has 2(j + 1) solutions
Using Sturm Comparison Theorem, we also can get a nonexistence result when f satisfies a nonresonance condition.
Theorem 1.4. Let (H_{3}) hold. Assume that there exists an integer k ∈ ℕ such that
for any t ∈ [0, 1], where λ_{k }denotes the kth eigenvalue of (1.5). Then BVP (1.4) has no nontrivial solution.
Remark 1.3. Similarly to Remark 1.2, we note that the assumptions (i) and (ii) are weaker than
the corresponding conditions of Theorem B. In fact, if we let f(t, s) ≡ w(t) f(s), then we can get
2 Preliminary results
To show the nodal solutions of the BVP (1.4), we need only consider an operator equation of the following form
Equations of the form (2.1) are usually called nonlinear eigenvalue problems. LópezGómez [7] studied a nonlinear eigenvalue problem of the form
where r ∈ ℝ is a parameter, u ∈ X, X is a Banach space, θ is the zero element of X, and G:
Lemma 2.1 [[7], Theorem 6.4.3]. Assume Σ(T) is discrete. Let λ_{0 }∈ Σ(T) such that ind(0, λ_{0}T) changes sign as λ crosses λ_{0}, then each of the components
(i) meets infinity in
(ii) meets (τ, θ), where τ ≠ λ_{0 }∈ Σ(T) or
(iii)
where V is the complement of
Lemma 2.2 [[7], Theorem 6.5.1]. Under the assumptions:
(A) X is an order Banach space, whose positive cone, denoted by P, is normal and has a nonempty interior;
(B) The family ϒ(r) has the special form
where T is a compact strongly positive operator, i.e., T(P\{0}) ⊂ int P;
(C) The solutions of u = rTu + H(r, u) satisfy the strong maximum principle.
Then the following assertions are true:
(1) Spr (T) is a simple eigenvalue of T, having a positive eigenfunction denoted by ψ_{0 }> 0, i.e., ψ_{0 }∈ int P, and there is no other eigenvalue of T with a positive eigenfunction;
(2) For every y ∈ int P, the equation
has exactly one positive solution if
Lemma 2.3 [[10], Theorem 2.5]. Assume T : X → X is a completely continuous linear operator, and 1 is not an eigenvalue of T, then
where β is the sum of the algebraic multiplicities of the eigenvalues of T large than 1, and β = 0 if T has no eigenvalue of this kind.
Let Y = C[0, 1] with the norm
with the norm
Define L: D(L) → Y by setting
where
Then L^{1}: Y → E is compact. Let
(i) u(0) = 0, νu'(0) > 0;
(ii) u has only simple zeros in [0, 1] and exactly n  1 zeros in (0,1).
Then sets
Furthermore, let ζ ∈ C[0, 1] × ℝ) be such that
with
Let
then
If u ∈ E, it follows from (2.3) that
uniformly for t ∈ [0, 1].
Let us study
as a bifurcation problem from the trivial solution u ≡ 0.
Equation (2.4) can be converted to the equivalent equation
Further we note that L^{1}[ζ(t, u(t))] _{E }= o(u_{E}) for u near 0 in E.
Lemma 2.4. For each k ∈ ℕ and ν ∈ {+.  }, there exists a continuum
(i)
(ii)
(iii)
Proof. It is easy to see that the problem (2.4) is of the form considered in [7], and satisfies the general hypotheses imposed in that article.
Combining Lemma 2.1 with Lemma 2.3, we know that there exists a continuum
(a)
(b) or
(c) or
where V is the complement of span{φ_{k}}, φ_{k }denotes the eigenfunction corresponding to eigenvalue λ_{k}.
We finally prove that the first choice of the (a) is the only possibility.
In fact, all functions belong to the continuum sets
Next, we shall prove (c) is impossible, suppose (c) occurs, then
Note that as the complement V of span{φ_{k}} in E, we can take
Thus, for this choice of V, the component
Indeed, if
then y > 0 in (0, a_{0}), where a_{0 }denotes the first zero point of y, and there exists u ∈ E for which
Thus, for each sufficiently large α > 0, we have that u + αφ_{k }>> 0 in (0, a_{0}) and
Define
Hence, according to Lemma 2.2
which is impossible since
Lemma 2.5. If
Proof. Taking into account Lemma 2.4, we only need to prove that
Suppose
By (2.3), (2.5) and the compactness of L^{1}, we obtain that for some convenient subsequence c_{j }→ c_{0 }≠ 0 as j → + ∞. Now c_{0 }verifies the equation
and c_{0}_{E }= 1. Hence μ* = λ_{i}, for some i ≠ k, i ∈ ℕ. Therefore, (μ_{j}, u_{j}) → (λ_{i}, θ) with
3 Proof of main results
Proof of Theorems 1.1 and 1.2. We only prove Theorem 1.1 since the proof of Theorem 1.2 is similar. It is clear
that any solution of (2.4) of the form (1, u) yields a solution u of (1.4). We shall show
By the strict decreasing of μ_{k}(c(t)) with respect to c(t) (see [11]), where μ_{k}(c(t)) is the kth eigenvalue of (1.2) corresponding to the weight function c(t), we have μ_{k}(c(t)) > μ_{k}(λ_{k}) = 1.
Let
We note that μ_{j }> 0 for all j ∈ ℕ, since (0,0) is the only solution of (2.4) for μ = 0 and
Step 1: We show that if there exists a constant M > 0, such that
for j ∈ ℕ large enough, then
In this case it follows that
Let ξ ∈ C([0, 1] × ℝ) be such that
with
We divide the equation
set
By the compactness of L we obtain
where
It is clear that
By the strict decreasing of
Step 2: We show that there exists a constant M such that μ_{j }∈ (0, M] for j ∈ ℕ large enough.
On the contrary, we suppose that
On the other hand, we note that
In view of Remark 1.1, we have
Therefore,
for some constant number M > 0 and j ∈ ℕ sufficiently large.
Proof of Theorem 1.3. Repeating the arguments used in the proof of Theorems 1.1 and 1.2, we see that for ν ∈ {+, } and each i ∈ {k, k + 1,..., k + j}
The results follows.
Proof of Theorem 1.4. Assume to the contrary that BVP (1.4) has a solution u ∈ E, we see that u satisfies
where
Note that
We know that the eigenfunction φ_{k }corresponding to λ_{k }has exactly k  1 zeros in [0, 1]. Applying Lemma 2.4 of [13] to φ_{k }and u, we see that u has at least k zeros in I. By Lemma 2.4 of [13] again to u and φ_{k+1}, we get that φ_{k+1 }has at least k + 1 zeros in [0, 1]. This is a contradiction.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
GD conceived of the study, and participated in its design and coordination and helped to draft the manuscript. BY drafted the manuscript. RM participated in the design of the study. All authors read and approved the final manuscript.
Acknowledgements
The authors were very grateful to the anonymous referees for their valuable suggestions. This study was supported by the NSFC (No. 11061030, No. 10971087) and NWNULKQN1021.
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