Nodal solutions of second-order two-point boundary value problems

We shall study the existence and multiplicity of nodal solutions of the nonlinear second-order two-point boundary value problems,

The proof of our main results is based upon bifurcation techniques.

Mathematics Subject Classifications: 34B07; 34C10; 34C23.

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 f₀, 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) f₀ < μ_k < ⋯ < μ_k+j < f_∞;

(ii) f_∞ < μ_k < ⋯ < μ_k+j < f₀,

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:

(H₁) 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

(H₂) 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);

(H₃) f(t, s)s > 0 for t ∈ (0, 1) and s ≠ 0.

Remark 1.1. From (H₁)-(H₃), 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 López-Gómez [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, López-Gómez gave a corrected version of unilateral bifurcation theorem in [7].

By applying the bifurcation theorem of López-Gómez [[7], Theorem 6.4.3], we shall establish the following:

Theorem 1.1. Suppose that f(t, u) satisfies (H₁), (H₂), and (H₃), 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 (H₃) 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 (H₁) and (H₂) 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 (H₃) 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 non-existence result when f satisfies a non-resonance condition.

Theorem 1.4. Let (H₃) 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 f₀ < 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 < f₀. Therefore, Theorem B is the corollary of Theorem 1.3. Similar, we get Theorem C is also the corollary of Theorem 1.4.

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. López-Gómez [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. López-Gómez [7] established the following results:

Lemma 2.1 [[7], Theorem 6.4.3]. Assume Σ(T) is discrete. Let λ₀ ∈ Σ(T) such that ind(0, λ₀T) changes sign as λ crosses λ₀, then each of the components satisfies , and either

(i) meets infinity in ,

(ii) meets (τ, θ), where τ ≠ λ₀ ∈ Σ(T) or

(iii) contains a point

where V is the complement of denotes the eigenfunction corresponding to eigenvalue λ₀.

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 ψ₀ > 0, i.e., ψ₀ ∈ 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 C¹ function u, if u(x₀) = 0, then x₀ is a simple zero of u, if u'(x₀) ≠ 0. For any integer k ∈ ℕ and ν ∈ {+, --}, define consisting of functions u ∈ C¹ [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:

(i) ;

(ii) ;

(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:

(a) is unbounded and ;

(b) or , where j ∈ ℕ, λ_j is another eigenvalue of (1.5) and different from λ_k;

(c) or contains a point

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, a₀), where a₀ 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, a₀) and

Define

Hence, according to Lemma 2.2

which is impossible since .

Lemma 2.5. If is a non-trivial 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 → c₀ ≠ 0 as j → + ∞. Now c₀ verifies the equation

and ||c₀||_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.

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.

Let with satisfies

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 C² [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

where .

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.

The authors declare that they have no competing interests.

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.

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 NWNU-LKQN-10-21.

1. Ma, R, Thompson, B: Nodal solutions for nonlinear eigenvalue problems. Nonlinear Anal TMA. 59, 707–718 (2004)

2. Walter, W: Ordinary Differential Equations. Springer, New York (1998)

3. Ma, R, Thompson, B: Multiplicity results for second-order two-point boundary value problems with nonlinearities across several eigenvalues. J Appl Math Lett. 18, 587–595 (2005). Publisher Full Text

4. Rabinowitz, PH: Some global results for nonlinear eigenvalue problems. J Funct Anal. 7, 487–513 (1971). Publisher Full Text

5. Dancer, EN: On the structure of solutions of non-linear eigenvalue problems. Indiana U Math J. 23, 1069–1076 (1974). Publisher Full Text

6. Dancer, EN: Bifurcation from simple eigenvaluses and eigenvalues of geometric multiplicity one. Bull Lond Math Soc. 34, 533–538 (2002). Publisher Full Text

7. López-Gómez, J: Spectral theory and nonlinear functional analysis. Chapman and Hall/CRC, Boca Raton (2001)

8. Blat, J, Brown, KJ: Bifurcation of steady state solutions in predator prey and competition systems. Proc Roy Soc Edinburgh. 97A, 21–34 (1984)

9. López-Gómez, J: Nonlinear eigenvalues and global bifurcation theory, application to the search of positive solutions for general Lotka-Volterra reaction diffusion systems. Diff Int Equ. 7, 1427–1452 (1984)

10. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic press, New York (1988)

11. Anane, A, Chakrone, O, Monssa, M: Spectrum of one dimensionalp-Laplacian with indefinite weight. Electron J Qual Theory Diff Equ. 2002(17), 11 (2002)

12. Dai, G, Ma, R: Unilateral global bifurcation and radial nodal solutions for the p-Laplacian in unit ball.

13. Lee, YH, Sim, I: Existence results of sign-changing solutions for singular one-dimensional p-Laplacian problems. Nonlinear Anal TMA. 68, 1195–1209 (2008). Publisher Full Text