Abstract
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 or . Then BVP (1.1) has two solutions and such that has exactly k  1 zeros in (0, 1) and is positive near 0, and has exactly k  1 zeros in (0,1) and is negative near 0.
In [3], Ma and Thompson studied the existence and multiplicity of nodal solutions for BVP
They gave conditions on the ratio at infinity and zero that guarantee the existence of solutions with prescribed nodal properties.
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 , such that has exactly k + i  1 zeros in (0, 1) and are positive near 0, and has exactly k + i  1 zeros in (0,1) and are negative near 0.
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}) uniformly on [0, 1], and the inequality is strict on some subset of positive measure in (0,1), where λ_{k }denotes the kth eigenvalue of
(H_{2}) uniformly on [0, 1], and all the inequalities are strict on some subset of positive measure in (0, 1), where λ_{k }denotes the kth eigenvalue of (1.5);
(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 for all r ∈ [α, β] and s ≠ 0.
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 and , such that has exactly k  1 zeros in (0, 1) and is positive near 0, and has exactly k  1 zeros in (0,1) and is negative near 0.
Similarly, we also have the following:
Theorem 1.2. Suppose that f(t, u) satisfies (H_{3}) and
uniformly on [0, 1], and all the inequalities are strict on some subset of positive measure in (0, 1), where λ_{k }denotes the kth eigenvalue of (1.5);
uniformly on [0, 1], and the inequality is strict on some subset of positive measure in (0, 1), where λ_{k }denotes the kth eigenvalue of (1.5), then problem (1.4) possesses two solutions and , such that has exactly k  1 zeros in (0,1) and is positive near 0, and has exactly k  1 zeros in (0,1) and is negative near 0.
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 and . By the strict decreasing of μ_{k}(f) with respect to weight function f (see [10]), where μ_{k}(f) denotes the kth eigenvalue of (1.2) corresponding to weight function f, we can show that our condition c(t) ≤ λ_{k }≤ a(t) is equivalent to the condition . Similarly, our condition c (t) ≥ λ_{k }≥ a (t) is equivalent to the condition . Therefore, Theorem A is the corollary of Theorems 1.1 and 1.2.
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) uniformly on [0, 1], and the inequalities are strict on some subset of positive measure in (0,1), where λ_{k }denotes the kth eigenvalue of (1.5);
(ii) uniformly on [0, 1], and the inequality is strict on some subset of positive measure in (0, 1), where λ_{k }denotes the kth eigenvalue of (1.5).
Then BVP (1.4) has 2(j + 1) solutions , such that has exactly k + i  1 zeros in (0,1) and are positive near 0, and has exactly k + i  1 zeros in (0,1) and are negative near 0.
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 and . By the strict decreasing of μ_{k}(f) with respect to weight function f (see [11]), where μ_{k}(f) denotes the kth eigenvalue of (1.2) corresponding to weight function f, we can show that our condition c(t) ≤ λ_{k }< ⋯ < λ_{k+j }≤ a(t) is equivalent to the condition f_{0 }< μ_{k }< · · · < μ_{k+j }< f_{∞}. Similarly, our condition a(t) ≤ λ_{k }< · · · < λ_{k+j }≤ c(t) is equivalent to the condition f_{∞ }< μ_{k }< ⋯ < μ_{k+j }< f_{0}. Therefore, Theorem B is the corollary of Theorem 1.3. Similar, we get Theorem C is also the corollary of Theorem 1.4.
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: is completely continuous. In addition, G(r, u) = rTu + H(r, u), where H(r, u) = o(u) as u → 0 uniformly on bounded r interval, and T is a linear completely continuous operator on X. A solution of (2.2) is a pair , which satisfies the equation (2.2). The closure of the set nontrivial solutions of (2.2) is denoted by ℂ, let Σ(T) denote the set of eigenvalues of linear operator T. LópezGómez [7] established the following results:
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 satisfies , and either
(ii) meets (τ, θ), where τ ≠ λ_{0 }∈ Σ(T) or
where V is the complement of denotes the eigenfunction corresponding to eigenvalue λ_{0}.
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 , whereas it does not admit a 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 . Let
with the norm
Define L: D(L) → Y by setting
where
Then L^{1}: Y → E is compact. Let under the product topology. 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. For any integer k ∈ ℕ and ν ∈ {+, }, define consisting of functions u ∈ C^{1 }[0, 1] satisfying the following conditions:
(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 are disjoint and open in E. Finally, let .
Furthermore, let ζ ∈ C[0, 1] × ℝ) be such that
with
Let
then is nondecreasing with respect to u and
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 of solutions of (2.4) with the properties:
(iii) is unbounded in , where λ_{k }denotes the kth eigenvalue of (1.5).
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 of solutions of (2.4) such that:
(b) or , where j ∈ ℕ, λ_{j }is another eigenvalue of (1.5) and different from λ_{k};
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 have exactly k  1 simple zeros, this implies that it is impossible to exist .
Next, we shall prove (c) is impossible, suppose (c) occurs, then is bounded and without loss of generality, suppose there exists a point . Moreover, it follows from Lemma 2.1 that
Note that as the complement V of span{φ_{k}} in E, we can take
Thus, for this choice of V, the component cannot contain a point
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
Lemma 2.5. If is a nontrivial solution of (2.4), then for ν and some k ∈ ℕ.
Proof. Taking into account Lemma 2.4, we only need to prove that .
Suppose . Then there exists such that , and (μ_{j},u_{j}) → (μ*, u) with . Since , so u ≡ 0. Let , then c_{j }should be a solution of problem,
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 . This contradicts to Lemma 2.3.
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 crosses the hyperplane {1} × E in ℝ × E.
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.
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 crosses the hyperplane {1} × E in ℝ × E.
In this case it follows that
Let ξ ∈ C([0, 1] × ℝ) be such that
with
We divide the equation
set . Since is bounded in C^{2 }[0, 1], after taking a subsequence if necessary, we have that for some with u_{E }= 1. By (3.1), using the similar proof of (2.3), we have that
By the compactness of L we obtain
where , again choosing a subsequence and relabeling if necessary.
It is clear that since is closed in ℝ × E. Therefore, is the kth eigenvalue of
By the strict decreasing of with respect to a(t) (see [11]), where is the kth eigenvalue of (1.2) corresponding to the weight function a(t), we have . Therefore, crosses the hyperplane {1} × E in ℝ × E.
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 on [α, β] and for j large enough and all t ∈ [0, 1]. By Lemma 3.2 of [12], we get u_{j }must change its sign more than k times on [α, β] for j large enough, which contradicts the act that .
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
Note that and hence can be regarded as a continuous function on ℝ. Thus we get b(·) ∈ C[0, 1]. Also, notice that a nontrivial solution of (1.4) has a finite number of zeros. From (2.8) and the above fact λ_{k }< b(t) < λ_{k+1 }for all t ∈ [0, 1].
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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