The first nontrivial curve in the fučĺk spectrum of the dirichlet laplacian on the ball consists of nonradial eigenvalues

  • Jiřĺ Benedikt1Email author,

    Affiliated with

    • Pavel Drábek2 and

      Affiliated with

      • Petr Girg1

        Affiliated with

        Boundary Value Problems20112011:27

        DOI: 10.1186/1687-2770-2011-27

        Received: 3 May 2011

        Accepted: 4 October 2011

        Published: 4 October 2011

        Abstract

        It is well-known that the second eigenvalue λ2 of the Dirichlet Laplacian on the ball is not radial. Recently, Bartsch, Weth and Willem proved that the same conclusion holds true for the so-called nontrivial (sign changing) Fučík eigenvalues on the first curve of the Fučík spectrum which are close to the point (λ2, λ2). We show that the same conclusion is true in dimensions 2 and 3 without the last restriction.

        Keywords

        Fučík spectrum The first curve of the Fučík spectrum Radial and nonradial eigenfunctions

        1. Introduction

        Let Ω ⊂ ℝ N be a bounded domain, N ≥ 2. The Fučík spectrum of -Δ on W 0 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq1_HTML.gif is defined as a set Σ of those (λ+, λ - ) ∈ ℝ2 such that the Dirichlet problem
        - Δ u = λ + u + - λ - u - i n Ω , u = 0 o n Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ1_HTML.gif
        (1)
        has a nontrivial solution u W 0 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq2_HTML.gif. In particular, if λ1< λ2< ⋯ are the eigenvalues of the Dirichlet Laplacian on Ω (counted with multiplicity), then clearly Σ contains each pair (λ k , λ k ), k ∈ ℕ, and the two lines {λ1} × ℝ and ℝ × {λ1}. Following [1, p. 15], we call the elements of Σ \ ({λ1} × ℝ ∪ ℝ × {λ1}) nontrivial Fučík eigenvalues. It was proved in [2] that there exists a first curve C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq3_HTML.gif of nontrivial Fučík eigenvalues in the sense that, defining η: (λ1, ∞) → ℝ by
        η ( λ ) def ¯ ¯ inf { μ > λ 1 : ( λ , μ ) is a nontrivial Fu č í k eigenvalue } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equa_HTML.gif
        we have that λ1< η(λ) < ∞ for every λ (1), and the curve
        C d e f ( λ , η ( λ ) ) : λ ( λ 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equb_HTML.gif

        consists of nontrivial Fučík eigenvalues. Moreover, it was proved in [2] that C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq3_HTML.gif is a continuous and strictly decreasing curve which contains the point (λ2, λ2) and which is symmetric with respect to the diagonal.

        It was conjectured in [1, p. 16], that if Ω is a radially symmetric bounded domain, then every eigenfunction u of (1) corresponding to some ( λ + , λ - ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq4_HTML.gifis not radial. The authors of [1, p. 16] actually proved that the conjecture is true if ( λ + , λ - ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq4_HTML.gifbut sufficiently close to the diagonal.

        The original purpose of this paper was to prove that the above conjecture holds true for all ( λ + , λ - ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq4_HTML.gif provided Ω is a ball in ℝ N with N = 2 and N = 3. Without loss of generality, we prove it for the unit ball B centred at the origin. Cf. Theorem 6 below.

        During the review of this paper, one of the reviewers drew the authors' attention to the paper [3], where the same result is proved for general N ≥ 2 (see [3, Theorem 3.2]). The proof in [3] uses the Morse index theory and covers also problems with weights on more general domains than balls. On the other hand, our proof is more elementary and geometrically instructive. From this point of view, our result represents a constructive alternative to the rather abstract approach presented in [3]. This is the main authors' contribution.

        2. Variational characterization of C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq3_HTML.gif

        Let us fix s ∈ ℝ and let us draw in the (λ+, λ-) plane a line parallel to the diagonal and passing through the point (s, 0), see Figure 1.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Fig1_HTML.jpg
        Figure 1

        The first two Fučík curves.

        We show that the point of intersection of this line and C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq3_HTML.gif corresponds to the critical value of some constrained functional (cf. [4, p. 214]). To this end we define the functional
        J s ( u ) d e f Ω | u | 2 - s Ω ( u + ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equc_HTML.gif
        Then J s ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq5_HTML.gif is a C1-functional on W 0 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq1_HTML.gif and we look for the critical points of the restriction J ̃ s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq6_HTML.gif of J s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq7_HTML.gif to
        S d e f u W 0 1 , 2 ( Ω ) : I ( u ) d e f Ω u 2 = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equd_HTML.gif
        By the Lagrange multipliers rule, u S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq8_HTML.gif is a critical point of J ̃ s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq6_HTML.gif if and only if there exists t ∈ ℝ such that
        J s ( u ) = t I ( u ) , i . e . , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Eque_HTML.gif
        Ω u v - s Ω u + v = t Ω u v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ2_HTML.gif
        (2)
        for all v W 0 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq9_HTML.gif. This means that
        - Δ u = ( s + t ) u + - t u - in  Ω , u = 0 on  Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equf_HTML.gif

        holds in the weak sense. In particular, (λ+, λ - ) = (s + t, t) ∈ Σ. Taking v = u in (2), one can see that the Lagrange multiplier t is equal to the corresponding critical value of J ̃ s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq6_HTML.gif.

        From now on we assume s ≥ 0, which is no restriction since Σ is clearly symmetric with respect to the diagonal. The first eigenvalue λ1 of - Δ on W 0 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq1_HTML.gif is defined as
        λ 1 = λ 1 ( Ω ) d e f min Ω | u | : u W 0 1 , 2 ( Ω ) and Ω | u | 2 = 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ3_HTML.gif
        (3)
        It is well known that λ1> 0, simple and admits an eigenfunction φ 1 W 0 1 , 2 ( Ω ) C 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq10_HTML.gif with φ1 satisfying φ1(x) > 0 for x ∈ Ω. Let
        Γ d e f γ C ( [ - 1 , 1 ] , S ) : γ ( - 1 ) = - φ 1 and γ ( 1 ) = φ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equg_HTML.gif
        and
        c ( s ) d e f inf γ Γ max u γ J ̃ s ( u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ4_HTML.gif
        (4)
        We keep the same notation γ for the image of a function γ = γ (t). It follows from [4, Props. 2.2, 2.3 and Thms. 2.10, 3.1] that the first three critical levels of J ̃ s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq6_HTML.gif are classified as follows.
        1. (i)

          φ 1 is a strict global minimum of J ̃ s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq6_HTML.gif with J ̃ s ( φ 1 ) = λ 1 - s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq11_HTML.gif. The corresponding point in Σ is (λ 1, λ 1 - s), which lies on the vertical line through (λ 1, λ 1).

           
        2. (ii)

          -φ 1 is a strict local minimum of J ̃ s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq6_HTML.gif, and J ̃ s ( - φ 1 ) = λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq12_HTML.gif. The corresponding point in Σ is (λ 1 + s, λ 1), which lies on the horizontal line through (λ 1, λ 1).

           
        3. (iii)

          For each s ≥ 0, the point (s + c(s), c(s)), where c(s) > λ 1 is defined by the minimax formula (4), belongs to Σ. Moreover, the point (s + c(s), c(s)) is the first nontrivial point of Σ on the parallel to the diagonal through (s, 0).

           

        Next we summarize some properties of the dependence of the (principal) first eigenvalue λ1(Ω) on the domain Ω. The following proposition follows immediately from the variational characterization of λ1 given by (3) and the properties of the corresponding eigenfunction φ1.

        Proposition 1. λ12) < λ11) whenever Ω i , i = 1, 2, are bounded domains satisfying Ω1 ⊆ Ω2and meas(Ω1) < meas(Ω2).

        Let us denote by V d , d ∈ (0, 1), the ball canopy of the height 2d and by B d the maximal inscribed ball in V d (see Figure 2). It follows from Proposition 1 that for d ∈ (0, 1), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Fig2_HTML.jpg
        Figure 2

        The ball decomposition

        λ 1 ( V d ) < λ 1 ( B d ) , λ 1 ( V 1 - d ) < λ 1 ( B 1 - d ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ5_HTML.gif
        (5)
        Moreover, from the variational characterization (3), the following properties of the function
        d λ 1 ( V d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ6_HTML.gif
        (6)

        follow immediately.

        Proposition 2. The function (6) is continuous and strictly decreasing on (0, 1), it maps (0, 1) onto (λ1(B), ∞) and lim d 0 + λ 1 ( V d ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq13_HTML.gif, lim d 1 - λ 1 ( V d ) = λ 1 ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq14_HTML.gif.

        In particular, it follows from Proposition 2 that, given s ≥ 0, there exists a unique d s ( 0 , 1 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq15_HTML.gif such that
        λ 1 ( V d s ) = s + λ 1 ( V 1 - d s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ7_HTML.gif
        (7)
        Let u d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq16_HTML.gif and u 1 - d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq17_HTML.gif be positive principle eigenvalues associated with λ 1 ( V d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq18_HTML.gif and λ 1 ( V 1 - d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq19_HTML.gif, respectively. We extend both functions on the entire B by setting u d s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq20_HTML.gif on V 1 - d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq21_HTML.gif, u 1 - d s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq22_HTML.gif on V d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq23_HTML.gif and then normalize them by u d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq16_HTML.gif, u 1 - d s S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq24_HTML.gif. Our aim is to construct a special curve γ ∈ Γ on which the values of J ̃ s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq6_HTML.gif stay below λ 1 ( V d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq18_HTML.gif. Actually, the curve γ connects φ1 with (1) and passes through u d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq16_HTML.gif and ( - u 1 - d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq25_HTML.gif. For this purpose we set γ = γ1γ2γ3, where
        γ 1 d e f u = ( τ φ 1 2 + ( 1 - τ ) u d s 2 ) 1 2 : τ [ 0 , 1 ] , γ 2 d e f u = α u d s - β u 1 - d s : α 0 , β 0 , α 2 + β 2 = 1 , γ 3 d e f u = - ( τ φ 1 2 + ( 1 - τ ) u 1 - d s 2 ) 1 2 : τ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equh_HTML.gif

        Changing suitably the parametrization of γ i , i = 1, 2, 3 (we skip the details for the brevity), γ can be viewed as a graph of a continuous function, mapping [-1, 1] into S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq26_HTML.gif. We prove

        Proposition 3. J ̃ s ( u ) λ 1 ( V 1 - d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq27_HTML.giffor all uγ.

        For the proof we need so-called ray-strict convexity of the functional
        J ( v ) d e f Ω v 1 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ8_HTML.gif
        (8)
        defined on
        V + d e f v : Ω ( 0 , ) : v 1 2 W 0 1 , 2 ( Ω ) C ( Ω ̄ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equi_HTML.gif
        We say that J : V + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq28_HTML.gif is ray-strictly convex if for all τ ∈ (0, 1) and v1, v2V+ we have
        J ( ( 1 - τ ) v 1 + τ v 2 ) ( 1 - τ ) J ( v 1 ) + τ J ( v 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equj_HTML.gif

        where the equality holds if and only if v1 and v2 are colinear.

        Lemma 4 (see [5, p. 132]). The functional J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq29_HTML.gifdefined by (8) is ray-strictly convex.

        Proof of Proposition 3.
        1. 1.
          The values on γ 1. For uγ 1 we have
          J ̃ s ( u ) = J ( u 2 ) - s B u 2 = B τ φ 1 2 + ( 1 - τ ) u d s 2 1 2 2 - s B τ φ 1 2 + ( 1 - τ ) u d s 2 τ B | φ 1 | 2 + ( 1 - τ ) B | u d s | 2 - s τ B φ 1 2 + ( 1 - τ ) B u d s 2 τ B | u d s | 2 + ( 1 - τ ) B | u d s | 2 - s V d s | u d s | 2 - s = λ 1 ( V d s ) - s = s + λ 1 ( V 1 - d s ) - s = λ 1 ( V 1 - d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equk_HTML.gif
           
        by Lemma 4 (with Ω := B), (3) and (7).
        1. 2.
          The values on γ 2. Let uγ 2, then there exist α ≥ 0, β ≥ 0, α 2 + β 2 = 1 and such that u = α u d s - β u 1 - d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq30_HTML.gif. Since the supports of u d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq16_HTML.gif and u 1 - d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq17_HTML.gif are mutually disjoint, we have
          J ̃ s ( u ) = α 2 V d s | u d s | 2 + β 2 V 1 - d s | u 1 - d s | 2 - α 2 s V d s u d s 2 = α 2 λ 1 ( V d s ) + β 2 λ 1 ( V 1 - d s ) - α 2 s = α 2 s + ( α 2 + β 2 ) λ 1 ( V 1 - d s ) - α 2 s = λ 1 ( V 1 - d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equl_HTML.gif
           
        by (7).
        1. 3.
          The values on γ 3. For uγ 3 we have (similarly as in the first case)
          J ̃ s ( u ) = B τ φ 1 2 + ( 1 - τ ) u 1 - d s 2 1 2 2 V 1 - d s | u 1 - d s | 2 = λ 1 ( V 1 - d s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equm_HTML.gif
           

        From Proposition 3, (4) and (5) we immediately get

        Proposition 5. Given s ≥ 0, we have
        c ( s ) λ 1 ( V 1 - d s ) < λ 1 ( B 1 - d s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ9_HTML.gif
        (9)

        3. Radial eigenfunctions

        Radial Fučík spectrum has been studied in [6]. Let |x| be the Euclidean norm of x ∈ ℝ N and u = u(|x|) be a radial solution of the problem
        - Δ u = λ + u + - λ - u - in  B , u = 0 on  B . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ10_HTML.gif
        (10)
        Set r = |x| and write v(r) = u(|x|). It follows from the regularity theory that (10) is equivalent to the singular problem
        v + N - 1 r v + λ + v + - λ - v - = 0 in  ( 0 , 1 ) , v ( 0 ) = 0 , v ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ11_HTML.gif
        (11)
        The authors of [6] provide a detailed characterization of the Fučík spectrum of (11) by means of the analysis of the linear equation associated to (11):
        v + N - 1 r v + λ v = 0 in  ( 0 , ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ12_HTML.gif
        (12)
        The function v is a solution of (12) if and only if v ^ ( r ) = r 1 2 ( N - 1 ) v ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq31_HTML.gif is a solution of
        v ^ + λ + ( N - 1 ) ( 3 - N ) 4 r 2 v ^ = 0 in  ( 0 , ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ13_HTML.gif
        (13)

        Note that the functions v and v ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq32_HTML.gif have the same zeros.

        Let us investigate the radial Fučík eigenvalues which lie on the line parallel to the diagonal and which passes through the point (s, 0) in the (λ+, λ- )-plane. The first two intersections coincide with the points (λ1, λ1 - s) and (λ1 + s, λ1). This fact follows from the radial symmetry of the principal eigenfunction of the Dirichlet Laplacian on the ball. A normalized radial eigenfunction associated with the next intersection has exactly two nodal domains and it is either positive or else negative at the origin. Let us denote the former eigenfunction by u1 and the latter one by u2, respectively. Let (λ1 + s, λ1) and (λ2 + s, λ2) be Fučík eigenvalues associated with u1 and u2, respectively. The property (iii) on page 5 implies that c(s) ≤ λ i , i = 1, 2.

        The main result of this paper states that the above inequalities are strict and it is formulated as follows.

        Theorem 6. Let N = 2 or N = 3 and s ∈ ℝ be arbitrary. Then
        c ( s ) < λ i , i = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equn_HTML.gif

        In particular, nontrivial Fučík eigenvalues on the first curve of the Fučík spectrum are not radial.

        Proof. Let u i (x) = v i (r), i = 1, 2, r = |x|. Then there exists d1 ∈ (0, 1) such that v1(r) is a solution of
        v + N - 1 r v + ( s + λ 1 ) v = 0 and v > 0 in  ( 0 , d 1 ) , v ( 0 ) = v ( d 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equo_HTML.gif
        and
        v + N - 1 r v + λ 1 v = 0 and v < 0 in  ( d 1 , 1 ) , v ( d 1 ) = v ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equp_HTML.gif
        After the substitution v ^ 1 ( r ) = r 1 2 ( N - 1 ) v 1 ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq33_HTML.gif, v ^ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq34_HTML.gif is a solution of
        v ^ + s + λ 1 + ( N - 1 ) ( 3 - N ) 4 r 2 v ^ = 0 and v ^ > 0 in  ( 0 , d 1 ) , v ^ ( 0 ) = v ^ ( d 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ14_HTML.gif
        (14)
        and
        v ^ + λ 1 + ( N - 1 ) ( 3 - N ) 4 r 2 v ^ = 0 and v ^ < 0 in  ( d 1 , 1 ) , v ^ ( d 1 ) = v ^ ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ15_HTML.gif
        (15)
        Let u1 = u1(x) and u2 = u2(x) be the principal positive eigenfunctions associated with λ 1 ( B d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq35_HTML.gif and λ 1 ( B 1 - d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq36_HTML.gif, respectively. Both u i , i = 1, 2, are radially symmetric with respect to the centre of the corresponding ball. Due to the invariance of the Laplace operator with respect to translations we may assume that both B d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq37_HTML.gif and B 1 - d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq38_HTML.gif are centred at the origin. We then set u i (x) = w i (r), i = 1, 2, r = |x|. The functions w i , i = 1, 2, solve
        w 1 + N - 1 r w 1 + λ 1 ( B d s ) w 1 = 0 and w 1 > 0 in  ( 0 , d s ) , w 1 ( 0 ) = w 1 ( d s ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equq_HTML.gif
        and
        w 2 + N - 1 r w 2 + λ 1 ( B 1 - d s ) w 2 = 0 and w 2 > 0 in  ( 0 , 1 - d s ) , w 2 ( 0 ) = w 2 ( 1 - d s ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equr_HTML.gif
        After the substitution ŵ i ( r ) = r 1 2 ( N - 1 ) w i ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq39_HTML.gif, i = 1, 2, we have
        ŵ 1 + λ 1 ( B d s ) + ( N - 1 ) ( 3 - N ) 4 r 2 ŵ 1 = 0 and ŵ 1 > 0 in  ( 0 , d s ) , ŵ 1 ( 0 ) = ŵ 1 ( d s ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ16_HTML.gif
        (16)
        and
        ŵ 2 + λ 1 ( B 1 - d s ) + ( N - 1 ) ( 3 - N ) 4 r 2 ŵ 2 = 0 and ŵ 2 > 0 in  ( 0 , 1 - d s ) , ŵ 2 ( 0 ) = ŵ 2 ( 1 - d s ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equs_HTML.gif
        The substitution ( r ) = - v ^ ( r + d 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq40_HTML.gif transforms (15) to
        + λ 1 + ( N - 1 ) ( 3 - N ) 4 ( r + d 1 ) 2 = 0 and > 0 in  ( 0 , 1 - d 1 ) , ( 0 ) = ( 1 - d 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ17_HTML.gif
        (17)
        Let us assume that λ 1 λ 1 ( V 1 - d s ) ( < λ 1 ( B 1 - d s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq41_HTML.gif and that d1> d s . Choose δ = d 1 - d s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq42_HTML.gif and set w ̃ 2 ( r ) = ŵ 2 ( r + δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq43_HTML.gif. Then w ̃ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq44_HTML.gif solves
        w ̃ 2 + λ 1 ( B 1 - d s ) + ( N - 1 ) ( 3 - N ) 4 ( r + δ ) 2 w ̃ 2 = 0 and w ̃ 2 > 0 in  ( - δ , 1 - d s - δ ) , w ̃ 2 ( - δ ) = w ̃ 2 ( 1 - d s - δ ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ18_HTML.gif
        (18)
        It follows that (18) is a Sturm majorant for (17) on the interval J = [ - δ 2 , 1 - d s - δ 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq45_HTML.gif and w ̃ 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq46_HTML.gif on J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq29_HTML.gif. Since ( 0 ) = ( 1 - d 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq47_HTML.gif and 0 J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq48_HTML.gif, 1 - d 1 J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq49_HTML.gif, we have a contradiction with the Sturm Separation Theorem (see [7, Cor. 3.1, p. 335]). Hence d1d s . Similar application of the Strum Separation Theorem to (14) and (16) now yields
        λ 1 ( B d s ) s + λ 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ19_HTML.gif
        (19)
        Since we also have λ 1 ( B d s ) > λ 1 ( V d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq50_HTML.gif, it follows from (7) and (19) that
        s + λ 1 ( V 1 - d s ) = λ 1 ( V d s ) < λ 1 ( B d s ) s + λ 1 s + λ 1 ( V 1 - d s ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equt_HTML.gif

        a contradiction which proves that λ 1 > λ 1 ( V 1 - d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq51_HTML.gif.

        Similarly as above, there exists d2 ∈ (0, 1) such that v2 is a solution of
        v + N - 1 r v + λ 2 v = 0 and v < 0 in  ( 0 , d 2 ) , v ( 0 ) = v ( d 2 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equu_HTML.gif
        and
        v + N - 1 r v + ( s + λ 2 ) v = 0 and v > 0 in  ( d 2 , 1 ) , v ( d 2 ) = v ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equv_HTML.gif
        After the substitution v ^ 2 ( r ) = r 1 2 ( N - 1 ) v 2 ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq52_HTML.gif, v ^ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq53_HTML.gif is a solution of
        v ^ + λ 2 + ( N - 1 ) ( 3 - N ) 4 r 2 v ^ = 0 and v ^ < 0 in  ( 0 , d 2 ) , v ^ ( 0 ) = v ^ ( d 2 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ20_HTML.gif
        (20)
        and
        v ^ + s + λ 2 + ( N - 1 ) ( 3 - N ) 4 r 2 v ^ = 0 and v ^ > 0 in  ( d 2 , 1 ) , v ^ ( d 2 ) = v ^ ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equ21_HTML.gif
        (21)
        Assume that λ 2 λ 1 ( V 1 - d s ) ( < λ 1 ( B 1 - d s ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq54_HTML.gif and that 1- d s > d2. Similar arguments based on the Sturm Comparison Theorem yield first that 1- d s d2 (i.e., 1 - d2d s ), and then (16), (21) that
        λ 1 ( B d s ) s + λ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equw_HTML.gif
        As above we obtain
        s + λ 1 ( V 1 - d s ) = λ 1 ( V d s ) < λ 1 ( B d s ) s + λ 2 s + λ 1 ( V 1 - d s ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_Equx_HTML.gif

        a contradiction which proves that λ 2 > λ 1 ( V 1 - d s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq55_HTML.gif.

        The assertion now follows from Proposition 5. ■

        Remark 7. Careful investigation of the above proof indicates that (N - 1)(3 - N) ≤ 0 is needed to make the comparison arguments work. The proof is simpler for N = 3 when the transformed equations for v ^ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq32_HTML.gif and ŵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-27/MediaObjects/13661_2011_Article_73_IEq56_HTML.gif are autonomous. The application of the Sturm Comparison Theorem is then more straightforward.

        Declarations

        6. Acknowledgments

        Jiří Benedikt and Petr Girg were supported by the Project KONTAKT, ME 10093, Pavel Drábek was supported by the Project KONTAKT, ME 09109.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia
        (2)
        Department of Mathematics and N.T.I.S., Faculty of Applied Sciences, University of West Bohemia

        References

        1. Bartsch T, Weth T, Willem M: Partial symmetry of least energy nodal solutions to some variational problems. J. D'Analyse Mathématique 2005, 96: 1-18.MathSciNetView Article
        2. de Figueiredo D, Gossez J-P: On the first curve of the Fučík spectrum of an elliptic operator. Differ. Integral Equ 1994, 7: 1285-1302.MathSciNet
        3. Bartsch T, Degiovanni M: Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains. Rend. Licei Mat. Appl 2006, 17: 69-85.MathSciNet
        4. Cuesta M, de Figueiredo D, Gossez J-P: The beginning of the Fučík spectrum for the p -Laplacian. J. Differ. Equ 1999, 159: 212-238. 10.1006/jdeq.1999.3645MathSciNetView Article
        5. Takáč P: Degenerate elliptic equations in ordered Banach spaces and applications. In Nonlinear Differential Equations. Chapman and Hall/CRC Res. Notes Math. Volume 404. Edited by: Drábek P, Krejčí P, Takáč P. CRC Press LLC, Boca Raton; 1999:111-196.
        6. Arias M, Campos J: Radial Fučik spectrum of the Laplace operator. J. Math. Anal. Appl 1995, 190: 654-666. 10.1006/jmaa.1995.1101MathSciNetView Article
        7. Hartman P: Ordinary Differential Equations. Wiley, New York; 1964.

        Copyright

        © Benedikt et al; licensee Springer. 2011

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