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

Jiřĺ Benedikt1*, Pavel Drábek2 and Petr Girg1

Author affiliations

1 Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia, Univerzitnĺ 22, 306 14 Plzeň, Czech Republic

2 Department of Mathematics and N.T.I.S., Faculty of Applied Sciences, University of West Bohemia, Univerzitnĺ 22, 306 14 Plzeň, Czech Republic

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Citation and License

Boundary Value Problems 2011, 2011:27  doi:10.1186/1687-2770-2011-27


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/27


Received:3 May 2011
Accepted:4 October 2011
Published:4 October 2011

© 2011 Benedikt et al; licensee Springer.

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.

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 ⊂ ℝN be a bounded domain, N ≥ 2. The Fučík spectrum of -Δ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M1">View MathML</a> is defined as a set Σ of those (λ+, λ-) ∈ ℝ2 such that the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M2">View MathML</a>

(1)

has a nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M3">View MathML</a>. In particular, if λ1 < λ2 < ⋯ are the eigenvalues of the Dirichlet Laplacian on Ω (counted with multiplicity), then clearly Σ contains each pair (λk, λk), 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M4">View MathML</a> of nontrivial Fučík eigenvalues in the sense that, defining η: (λ1, ∞) → ℝ by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M5">View MathML</a>

we have that λ1 < η(λ) < ∞ for every λ (1), and the curve

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M6">View MathML</a>

consists of nontrivial Fučík eigenvalues. Moreover, it was proved in [2] that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M4">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M7">View MathML</a>is not radial. The authors of [1, p. 16] actually proved that the conjecture is true if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M7">View MathML</a>but sufficiently close to the diagonal.

The original purpose of this paper was to prove that the above conjecture holds true for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M7">View MathML</a> provided Ω is a ball in ℝN 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M4">View MathML</a>

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.

thumbnailFigure 1. The first two Fučík curves.

We show that the point of intersection of this line and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M4">View MathML</a> corresponds to the critical value of some constrained functional (cf. [4, p. 214]). To this end we define the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M8">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M9">View MathML</a> is a C1-functional on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M1">View MathML</a> and we look for the critical points of the restriction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M11">View MathML</a> to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M12">View MathML</a>

By the Lagrange multipliers rule, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M13">View MathML</a> is a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10">View MathML</a> if and only if there exists t ∈ ℝ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M14">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M15">View MathML</a>

(2)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M16">View MathML</a>. This means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M17">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M1">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M18">View MathML</a>

(3)

It is well known that λ1 > 0, simple and admits an eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M19">View MathML</a> with φ1 satisfying φ1(x) > 0 for x ∈ Ω. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M20">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M21">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10">View MathML</a> are classified as follows.

(i) φ1 is a strict global minimum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M22">View MathML</a>. The corresponding point in Σ is (λ1, λ1 - s), which lies on the vertical line through (λ1, λ1).

(ii) -φ1 is a strict local minimum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M23">View MathML</a>. The corresponding point in Σ is (λ1 + s, λ1), which lies on the horizontal line through (λ1, λ1).

(iii) For each s ≥ 0, the point (s + c(s), c(s)), where c(s) > λ1 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 Ωi, i = 1, 2, are bounded domains satisfying Ω1 ⊆ Ω2 and meas(Ω1) < meas(Ω2).

Let us denote by Vd, d ∈ (0, 1), the ball canopy of the height 2d and by Bd the maximal inscribed ball in Vd (see Figure 2). It follows from Proposition 1 that for d ∈ (0, 1), we have

thumbnailFigure 2. The ball decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M24">View MathML</a>

(5)

Moreover, from the variational characterization (3), the following properties of the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M25">View MathML</a>

(6)

follow immediately.

Proposition 2. The function (6) is continuous and strictly decreasing on (0, 1), it maps (0, 1) onto (λ1(B), ∞) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M27">View MathML</a>.

In particular, it follows from Proposition 2 that, given s ≥ 0, there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M28">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M29">View MathML</a>

(7)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M31">View MathML</a> be positive principle eigenvalues associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M33">View MathML</a>, respectively. We extend both functions on the entire B by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M34">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M36">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M37">View MathML</a> and then normalize them by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M38">View MathML</a>. Our aim is to construct a special curve γ ∈ Γ on which the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M10">View MathML</a> stay below <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M32">View MathML</a>. Actually, the curve γ connects φ1 with (1) and passes through <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M39">View MathML</a>. For this purpose we set γ = γ1 γ2 γ3, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M40">View MathML</a>

Changing suitably the parametrization of γi, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M41">View MathML</a>. We prove

Proposition 3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M42">View MathML</a>for all u γ.

For the proof we need so-called ray-strict convexity of the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M43">View MathML</a>

(8)

defined on

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M44">View MathML</a>

We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M45">View MathML</a> is ray-strictly convex if for all τ ∈ (0, 1) and v1, v2 V+ we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M46">View MathML</a>

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

Lemma 4 (see [5, p. 132]). The functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M47">View MathML</a>defined by (8) is ray-strictly convex.

Proof of Proposition 3.

1. The values on γ1. For u γ1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M48">View MathML</a>

by Lemma 4 (with Ω := B), (3) and (7).

2. The values on γ2. Let u γ2, then there exist α ≥ 0, β ≥ 0, α2 + β2 = 1 and such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M49">View MathML</a>. Since the supports of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M31">View MathML</a> are mutually disjoint, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M50">View MathML</a>

by (7).

3. The values on γ3. For u γ3 we have (similarly as in the first case)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M51">View MathML</a>

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

Proposition 5. Given s ≥ 0, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M52">View MathML</a>

(9)

3. Radial eigenfunctions

Radial Fučík spectrum has been studied in [6]. Let |x| be the Euclidean norm of x ∈ ℝN and u = u(|x|) be a radial solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M53">View MathML</a>

(10)

Set r = |x| and write v(r) = u(|x|). It follows from the regularity theory that (10) is equivalent to the singular problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M54">View MathML</a>

(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):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M55">View MathML</a>

(12)

The function v is a solution of (12) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M56">View MathML</a> is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M57">View MathML</a>

(13)

Note that the functions v and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M58">View MathML</a> 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) ≤ λi, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M59">View MathML</a>

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

Proof. Let ui(x) = vi(r), i = 1, 2, r = |x|. Then there exists d1 ∈ (0, 1) such that v1(r) is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M60">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M61">View MathML</a>

After the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M63">View MathML</a> is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M64">View MathML</a>

(14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M65">View MathML</a>

(15)

Let u1 = u1(x) and u2 = u2(x) be the principal positive eigenfunctions associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M67">View MathML</a>, respectively. Both ui, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M69">View MathML</a> are centred at the origin. We then set ui(x) = wi(r), i = 1, 2, r = |x|. The functions wi, i = 1, 2, solve

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M70">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M71">View MathML</a>

After the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M72">View MathML</a>, i = 1, 2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M73">View MathML</a>

(16)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M74">View MathML</a>

The substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M75">View MathML</a> transforms (15) to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M76">View MathML</a>

(17)

Let us assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M77">View MathML</a> and that d1 > ds. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M78">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M79">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M80">View MathML</a> solves

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M81">View MathML</a>

(18)

It follows that (18) is a Sturm majorant for (17) on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M83">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M47">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M84">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M86">View MathML</a>, we have a contradiction with the Sturm Separation Theorem (see [7, Cor. 3.1, p. 335]). Hence d1 ds. Similar application of the Strum Separation Theorem to (14) and (16) now yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M87">View MathML</a>

(19)

Since we also have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M88">View MathML</a>, it follows from (7) and (19) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M89">View MathML</a>

a contradiction which proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M90">View MathML</a>.

Similarly as above, there exists d2 ∈ (0, 1) such that v2 is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M91">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M92">View MathML</a>

After the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M94">View MathML</a> is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M95">View MathML</a>

(20)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M96">View MathML</a>

(21)

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M97">View MathML</a> and that 1- ds > d2. Similar arguments based on the Sturm Comparison Theorem yield first that 1- ds d2 (i.e., 1 - d2 ds), and then (16), (21) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M98">View MathML</a>

As above we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M99">View MathML</a>

a contradiction which proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M100">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/27/mathml/M101">View MathML</a> are autonomous. The application of the Sturm Comparison Theorem is then more straightforward.

4. Competing interests

The authors declare that they have no competing interests.

5. Authors' contribution

All authors contributed to each part of this work equally.

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.

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