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

  • Ruyun Ma1,

    Affiliated with

    • Bianxia Yang1 and

      Affiliated with

      • Guowei Dai1Email author

        Affiliated with

        Boundary Value Problems20122012:13

        DOI: 10.1186/1687-2770-2012-13

        Received: 16 August 2011

        Accepted: 10 February 2012

        Published: 10 February 2012

        Abstract

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

        u + f ( t , u ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equa_HTML.gif

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

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

        Keywords

        nodal solutions bifurcation

        1 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)
        u ( t ) + μ w ( t ) f ( u ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ1_HTML.gif
        (1.1)

        under the assumptions:

        (C1) w(·) ∈ C([0, 1], [0, ∞)) and does not vanish identically on any subinterval of [0, 1];

        (C2) fC(ℝ, ℝ) with sf(s) > 0 for s ≠ 0;

        (C3) there exist f0, f∞ ∈ (0, ∞) such that
        f 0 = lim | s | 0 f ( s ) s , f = lim | s | f ( s ) s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equb_HTML.gif
        It is well known that under (C1) assumption, the eigenvalue problem
        φ ( t ) + μ w ( t ) φ ( t ) = 0 , t ( 0 , 1 ) , φ ( 0 ) = φ ( 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ2_HTML.gif
        (1.2)
        has a countable number of simple eigenvalues μ k , k = 1, 2,..., which satisfy
        0 < μ 1 < μ 2 < < μ k < , and lim k μ k = , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equc_HTML.gif

        and let μ k be the k th 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 μ k f < μ < μ k f 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq1_HTML.gif or μ k f 0 < μ < u k f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq2_HTML.gif. Then BVP (1.1) has two solutions u k + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq3_HTML.gif and u k - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq4_HTML.gif such that u k + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq3_HTML.gif has exactly k -- 1 zeros in (0, 1) and is positive near 0, and u k - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq4_HTML.gif 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
        u ( t ) + w ( t ) f ( u ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ3_HTML.gif
        (1.3)

        They gave conditions on the ratio f ( s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq5_HTML.gif 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 < ⋯ < μk+j< f;

        (ii) f < μ k < ⋯ < μk+j< f0,

        where μ k denotes the k th eigenvalue of (1.2). Then BVP (1.3) has 2(j + 1) solutions u k + i + , u k + i - , i = 0 , , j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq6_HTML.gif, such that u k + i + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq7_HTML.gif has exactly k + i -- 1 zeros in (0, 1) and are positive near 0, and u k + i - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq8_HTML.gif 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
        μ k - 1 < f ( s ) s < μ k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equd_HTML.gif

        where μ k denotes the k th 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
        u + f ( t , u ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ4_HTML.gif
        (1.4)

        under the following assumptions:

        (H1) λ k a ( t ) lim s + f ( t , s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq9_HTML.gif uniformly on [0, 1], and the inequality is strict on some subset of positive measure in (0,1), where λk denotes the k th eigenvalue of
        u ( t ) + λ u ( t ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ5_HTML.gif
        (1.5)

        (H2) 0 lim s 0 f ( t , s ) s c ( t ) λ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq10_HTML.gif uniformly on [0, 1], and all the inequalities are strict on some subset of positive measure in (0, 1), where λ k denotes the k th 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 f ( r , s ) s ϱ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq11_HTML.gif 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 u k + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq3_HTML.gif and u k - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq4_HTML.gif, such that u k + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq3_HTML.gif has exactly k -- 1 zeros in (0, 1) and is positive near 0, and u k - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq4_HTML.gif 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

        ( H 1 ) λ k a ( x ) lim s + f ( t , s ) s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq12_HTML.gif uniformly on [0, 1], and all the inequalities are strict on some subset of positive measure in (0, 1), where λk denotes the k th eigenvalue of (1.5);

        ( H 2 ) lim s 0 f ( t , s ) s c ( x ) λ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq13_HTML.gif uniformly on [0, 1], and the inequality is strict on some subset of positive measure in (0, 1), where λ k denotes the k th eigenvalue of (1.5), then problem (1.4) possesses two solutions u k + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq3_HTML.gif and u k - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq4_HTML.gif, such that u k + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq3_HTML.gif has exactly k-- 1 zeros in (0,1) and is positive near 0, and u k - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq4_HTML.gif 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 lim s + f ( t , s ) s μ w ( t ) f : = a ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq14_HTML.gif and lim s 0 f ( t , s ) s μ w ( t ) f 0 : = c ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq15_HTML.gif. By the strict decreasing of μ k (f) with respect to weight function f (see [10]), where μ k (f) denotes the k th 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 μ k f < μ < μ k f 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq1_HTML.gif. Similarly, our condition c (t) ≥ λka (t) is equivalent to the condition μ k f 0 < μ < u k f http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq2_HTML.gif. 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) 0 c ( t ) lim s 0 f ( t , s ) s λ k < < λ k + j a ( t ) lim s + f ( t , s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq16_HTML.gif uniformly on [0, 1], and the inequalities are strict on some subset of positive measure in (0,1), where λ k denotes the k th eigenvalue of (1.5);

        (ii) 0 a ( t ) lim s + f ( t , s ) s λ k < < λ k + j c ( t ) lim s + f ( t , s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq17_HTML.gif uniformly on [0, 1], and the inequality is strict on some subset of positive measure in (0, 1), where λ k denotes the k th eigenvalue of (1.5).

        Then BVP (1.4) has 2(j + 1) solutions u k + i + , u k + i - , i = 0 , , j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq6_HTML.gif, such that u k + i + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq7_HTML.gif has exactly k + i -- 1 zeros in (0,1) and are positive near 0, and u k + i - http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq8_HTML.gif 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
        λ k - 1 < f ( t , u ) u < λ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ6_HTML.gif
        (1.6)

        for any t ∈ [0, 1], where λ k denotes the k th 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 lim s + f ( t , s ) s w ( t ) f : = a ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq18_HTML.gif and lim s 0 f ( t , s ) s w ( t ) f 0 : = c ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq19_HTML.gif. By the strict decreasing of μ k (f) with respect to weight function f (see [11]), where μ k (f) denotes the k th eigenvalue of (1.2) corresponding to weight function f, we can show that our condition c(t) ≤ λ k < ⋯ < λk+ja(t) is equivalent to the condition f0 < μ k < · · · < μk+j< f. Similarly, our condition a(t) ≤ λ k < · · · < λk+jc(t) is equivalent to the condition f < μ k < ⋯ < μk+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.

        2 Preliminary results

        To show the nodal solutions of the BVP (1.4), we need only consider an operator equation of the following form
        u = λ A u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ7_HTML.gif
        (2.1)
        Equations of the form (2.1) are usually called nonlinear eigenvalue problems. López-Gómez [7] studied a nonlinear eigenvalue problem of the form
        u = G ( r , u ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ8_HTML.gif
        (2.2)

        where r ∈ ℝ is a parameter, uX, X is a Banach space, θ is the zero element of X, and G: X = × X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq20_HTML.gif 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 ( r , u ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq21_HTML.gif, 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 λ 0 ν , ν { + , - } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq22_HTML.gif satisfies ( λ 0 , θ ) λ 0 ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq23_HTML.gif, and either

        (i) meets infinity in X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq24_HTML.gif,

        (ii) meets (τ, θ), where τλ0 ∈ Σ(T) or

        (iii) λ 0 ν , ν { + , - } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq22_HTML.gif contains a point
        ( ι , y ) × ( V \ { 0 } ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Eque_HTML.gif

        where V is the complement of span { φ λ 0 } , φ λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq25_HTML.gif 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
        ϒ ( r ) = I X - r T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equf_HTML.gif

        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
        u - r T u = y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equg_HTML.gif

        has exactly one positive solution if r < 1 Spr ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq26_HTML.gif, whereas it does not admit a positive solution if r 1 Spr ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq27_HTML.gif.

        Lemma 2.3 [[10], Theorem 2.5]. Assume T : XX is a completely continuous linear operator, and 1 is not an eigenvalue of T, then
        i n d ( I - T , θ ) = ( - 1 ) β , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equh_HTML.gif

        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 u = max t [ 0 , 1 ] u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq28_HTML.gif. Let
        E = { u C 1 [ 0 , 1 ] | u ( 0 ) = u ( 1 ) = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equi_HTML.gif
        with the norm
        u E = max t [ 0 , 1 ] u + max t [ 0 , 1 ] u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equj_HTML.gif
        Define L: D(L) → Y by setting
        L u : = - u ( t ) , t [ 0 , 1 ] , u D ( L ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equk_HTML.gif
        where
        D ( L ) = { u C 2 [ 0 , 1 ] | u ( 0 ) = u ( 1 ) = 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equl_HTML.gif
        Then L-1: YE is compact. Let E = × E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq29_HTML.gif 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 S k ν C 1 [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq30_HTML.gif consisting of functions uC1 [0, 1] satisfying the following conditions:
        1. (i)

          u(0) = 0, νu'(0) > 0;

           
        2. (ii)

          u has only simple zeros in [0, 1] and exactly n -- 1 zeros in (0,1).

           

        Then sets S k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq31_HTML.gif are disjoint and open in E. Finally, let ϕ k ν = × S k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq32_HTML.gif.

        Furthermore, let ζ ∈ C[0, 1] × ℝ) be such that
        f ( t , u ) = c ( t ) u + ς ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equm_HTML.gif
        with
        lim u 0 ς ( t , u ) u = 0 and lim u ς ( t , u ) u = a ( t ) - c ( t ) uniformly on [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ9_HTML.gif
        (2.3)
        Let
        ς ̄ ( t , u ) = max 0 s u g ( t , u ) for t [ 0 , 1 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equn_HTML.gif
        then ς ̄ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq33_HTML.gif is nondecreasing with respect to u and
        lim u 0 + ς ̄ ( t , u ) u = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equo_HTML.gif
        If u ∈ E, it follows from (2.3) that
        ς ( t , u ) u E ς ̄ ( t , u ) u E ς ̄ ( t , u ) u E ς ̄ ( t , u E ) u E 0 , as u E 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equp_HTML.gif

        uniformly for t ∈ [0, 1].

        Let us study
        L u - μ c ( t ) u = μ ς ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ10_HTML.gif
        (2.4)

        as a bifurcation problem from the trivial solution u ≡ 0.

        Equation (2.4) can be converted to the equivalent equation
        u ( t ) = μ L - 1 [ c ( t ) u ( t ) ] + μ L - 1 [ ς ( t , u ( t ) ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equq_HTML.gif

        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 C k ν ϕ k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq34_HTML.gif of solutions of (2.4) with the properties:

        (i) ( λ k , θ ) C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq35_HTML.gif;

        (ii) C k ν \ { ( λ k , θ ) } ϕ k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq36_HTML.gif;

        (iii) C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq37_HTML.gif is unbounded in E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq38_HTML.gif, where λ k denotes the k th 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 C k ν E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq39_HTML.gif of solutions of (2.4) such that:
        1. (a)

          C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq40_HTML.gif is unbounded and ( λ k , θ ) C k ν , C k ν \ { ( λ k , θ ) } ϕ k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq41_HTML.gif;

           
        2. (b)

          or ( λ j , θ ) C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq42_HTML.gif, where j ∈ ℕ, λ j is another eigenvalue of (1.5) and different from λ k ;

           
        3. (c)
          or C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq37_HTML.gif contains a point
          ( ι , y ) × ( V \ { 0 } ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Eque_HTML.gif
           

        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 C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq40_HTML.gif have exactly k -- 1 simple zeros, this implies that it is impossible to exist ( λ j , θ ) C k ν , j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq43_HTML.gif.

        Next, we shall prove (c) is impossible, suppose (c) occurs, then C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq40_HTML.gif is bounded and without loss of generality, suppose there exists a point ( ι , y ) × ( V \ { θ } ) C k + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq44_HTML.gif. Moreover, it follows from Lemma 2.1 that
        C k + { ( λ , θ ) : λ } = { ( λ k , θ ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equr_HTML.gif
        Note that as the complement V of span{φ k } in E, we can take
        V : = R [ I E - λ k L ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equs_HTML.gif
        Thus, for this choice of V, the component C k + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq45_HTML.gif cannot contain a point
        ( ι , y ) × ( V \ { θ } ) C k + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equt_HTML.gif
        Indeed, if
        ( ι , y ) × ( V \ { θ } ) C k + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equt_HTML.gif
        then y > 0 in (0, a0), where a0 denotes the first zero point of y, and there exists uE for which
        u - λ k L u = y > 0 , in ( 0 , a 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equu_HTML.gif
        Thus, for each sufficiently large α > 0, we have that u + αφ k >> 0 in (0, a0) and
        u + α φ k - λ k L ( u + α φ k ) = y > 0 in ( 0 , a 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equv_HTML.gif
        Define
        P = { u E | u ( t ) 0 , t [ 0 , a 0 ] } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equw_HTML.gif
        Hence, according to Lemma 2.2
        Spr ( λ k L ) < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equx_HTML.gif

        which is impossible since Spr ( L ) = 1 λ k f 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq46_HTML.gif.

        Lemma 2.5. If ( μ , u ) E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq47_HTML.gif is a non-trivial solution of (2.4), then u S k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq48_HTML.gif for ν and some k ∈ ℕ.

        Proof. Taking into account Lemma 2.4, we only need to prove that C k ν Φ k ν { ( λ k , θ ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq49_HTML.gif.

        Suppose C k ν Φ k ν { ( λ k , θ ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq50_HTML.gif. Then there exists ( μ * , u ) C k ν ( × S k ν ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq51_HTML.gif such that ( μ * , u ) ( λ k , θ ) , u S k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq52_HTML.gif, and (μ j ,u j ) → (μ*, u) with ( μ j , u j ) C k ν ( × S k ν ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq53_HTML.gif. Since u S k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq54_HTML.gif, so u ≡ 0. Let c j : = u j u j E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq55_HTML.gif, then c j should be a solution of problem,
        c j = μ j L - 1 c ( t ) c j ( t ) + ς ( t , u j ( t ) ) u j E . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ11_HTML.gif
        (2.5)
        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
        - c 0 ( t ) = μ * c ( t ) c o ( t ) , t ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equy_HTML.gif

        and ||c0|| E = 1. Hence μ* = λ i , for some ik, i ∈ ℕ. Therefore, (μ j , u j ) → (λ i , θ) with ( μ j , u j ) C k ( × S k ν ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq56_HTML.gif. 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 C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq40_HTML.gif 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 k th eigenvalue of (1.2) corresponding to the weight function c(t), we have μ k (c(t)) > μ k (λ k ) = 1.

        Let ( μ j , u j ) C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq57_HTML.gif with u j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq58_HTML.gif satisfies
        μ j + u j E + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equz_HTML.gif

        We note that μ j > 0 for all j ∈ ℕ, since (0,0) is the only solution of (2.4) for μ = 0 and C k ν ( { 0 } × E ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq59_HTML.gif.

        Step 1: We show that if there exists a constant M > 0, such that
        μ j ( 0 , M ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equaa_HTML.gif

        for j ∈ ℕ large enough, then C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq40_HTML.gif crosses the hyperplane {1} × E in ℝ × E.

        In this case it follows that
        u j E . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equab_HTML.gif
        Let ξ ∈ C([0, 1] × ℝ) be such that
        f ( t , u ) = a ( t ) u + ξ ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equac_HTML.gif
        with
        lim u + ξ ( t , u ) u = 0 and lim u 0 ξ ( t , u ) u = c ( t ) - a ( t ) , uniformly on [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ12_HTML.gif
        (3.1)
        We divide the equation
        L u j - μ j a ( t ) u j = μ j ξ ( t , u j ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equ13_HTML.gif
        (3.2)
        set ū j = u j ū j E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq60_HTML.gif. Since ū j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq61_HTML.gif is bounded in C2 [0, 1], after taking a subsequence if necessary, we have that ū j ū http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq62_HTML.gif for some ū E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq63_HTML.gif with ||u|| E = 1. By (3.1), using the similar proof of (2.3), we have that
        lim j + ξ ( t , u j ( t ) ) u j E = 0 in Y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equad_HTML.gif
        By the compactness of L we obtain
        - ū - μ ̄ a ( t ) ū = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equae_HTML.gif

        where μ ̄ = lim j + μ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq64_HTML.gif, again choosing a subsequence and relabeling if necessary.

        It is clear that ū C k ν ¯ C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq65_HTML.gif since C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq66_HTML.gif is closed in ℝ × E. Therefore, μ ̄ ( a ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq67_HTML.gif is the k th eigenvalue of
        u ( t ) + μ a ( t ) u ( t ) = 0 , t ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equaf_HTML.gif

        By the strict decreasing of μ ̄ ( a ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq67_HTML.gif with respect to a(t) (see [11]), where μ ̄ ( a ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq67_HTML.gif is the k th eigenvalue of (1.2) corresponding to the weight function a(t), we have μ ̄ ( a ( t ) ) < μ ̄ ( λ k ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq68_HTML.gif. Therefore, C k ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq66_HTML.gif 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
        lim j + μ j = + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equag_HTML.gif
        On the other hand, we note that
        - u j = μ j f ( t , u j ) u j u j . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equah_HTML.gif

        In view of Remark 1.1, we have μ j f ( t , u j ) u j > λ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq69_HTML.gif 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 u j S k μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq70_HTML.gif.

        Therefore,
        μ j M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equai_HTML.gif

        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}
        C i ν ( { 1 } × E ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equaj_HTML.gif

        The results follows.

        Proof of Theorem 1.4. Assume to the contrary that BVP (1.4) has a solution uE, we see that u satisfies
        u ( t ) + b ( t ) u ( t ) = 0 , t ( 0 , 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_Equak_HTML.gif

        where b ( t ) = f ( t , u ) u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq71_HTML.gif.

        Note that c ( t ) lim s 0 f ( t , s ) s λ k + 1 < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq72_HTML.gif and hence f ( t , u ) u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-13/MediaObjects/13661_2011_Article_117_IEq73_HTML.gif 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) < λk+1for 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 φk+1, we get that φk+1 has at least k + 1 zeros in [0, 1]. This is a contradiction.

        Declarations

        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 NWNU-LKQN-10-21.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Northwest Normal University

        References

        1. Ma R, Thompson B: Nodal solutions for nonlinear eigenvalue problems. Nonlinear Anal TMA 2004, 59: 707–718.MathSciNetView Article
        2. Walter W: Ordinary Differential Equations. Springer, New York; 1998.View Article
        3. Ma R, Thompson B: Multiplicity results for second-order two-point boundary value problems with nonlinearities across several eigenvalues. J Appl Math Lett 2005, 18: 587–595. 10.1016/j.aml.2004.09.011MathSciNetView Article
        4. Rabinowitz PH: Some global results for nonlinear eigenvalue problems. J Funct Anal 1971, 7: 487–513. 10.1016/0022-1236(71)90030-9MathSciNetView Article
        5. Dancer EN: On the structure of solutions of non-linear eigenvalue problems. Indiana U Math J 1974, 23: 1069–1076. 10.1512/iumj.1974.23.23087MathSciNetView Article
        6. Dancer EN: Bifurcation from simple eigenvaluses and eigenvalues of geometric multiplicity one. Bull Lond Math Soc 2002, 34: 533–538. 10.1112/S002460930200108XMathSciNetView Article
        7. López-Gómez J: Spectral theory and nonlinear functional analysis. Chapman and Hall/CRC, Boca Raton; 2001.View Article
        8. Blat J, Brown KJ: Bifurcation of steady state solutions in predator prey and competition systems. Proc Roy Soc Edinburgh 1984, 97A: 21–34.MathSciNetView Article
        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 1984, 7: 1427–1452.
        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, 2002(17):11.
        12. Dai G, Ma R: Unilateral global bifurcation and radial nodal solutions for the p-Laplacian in unit ball. , in press.
        13. Lee YH, Sim I: Existence results of sign-changing solutions for singular one-dimensional p-Laplacian problems. Nonlinear Anal TMA 2008, 68: 1195–1209. 10.1016/j.na.2006.12.015MathSciNetView Article

        Copyright

        © Ma et al; licensee Springer. 2012

        This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.