The first nontrivial curve in the fučĺk spectrum of the dirichlet laplacian on the ball consists of nonradial eigenvalues
© Benedikt et al; licensee Springer. 2011
Received: 3 May 2011
Accepted: 4 October 2011
Published: 4 October 2011
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.
consists of nontrivial Fučík eigenvalues. Moreover, it was proved in  that 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 is not radial. The authors of [1, p. 16] actually proved that the conjecture is true if but sufficiently close to the diagonal.
The original purpose of this paper was to prove that the above conjecture holds true for all 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 , where the same result is proved for general N ≥ 2 (see [3, Theorem 3.2]). The proof in  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 . This is the main authors' contribution.
2. Variational characterization of
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 .
φ 1 is a strict global minimum of with . The corresponding point in Σ is (λ 1, λ 1 - s), which lies on the vertical line through (λ 1, λ 1).
-φ 1 is a strict local minimum of , and . The corresponding point in Σ is (λ 1 + s, λ 1), which lies on the horizontal line through (λ 1, λ 1).
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. λ1(Ω2) < λ1(Ω1) whenever Ω i , i = 1, 2, are bounded domains satisfying Ω1 ⊆ Ω2and meas(Ω1) < meas(Ω2).
Proposition 2. The function (6) is continuous and strictly decreasing on (0, 1), it maps (0, 1) onto (λ1(B), ∞) and , .
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 . We prove
Proposition 3. for all u ∈ γ.
where the equality holds if and only if v1 and v2 are colinear.
Lemma 4 (see [5, p. 132]). The functionaldefined by (8) is ray-strictly convex.
- 1.The values on γ 1. For u ∈ γ 1 we have
- 2.The values on γ 2. Let u ∈ γ 2, then there exist α ≥ 0, β ≥ 0, α 2 + β 2 = 1 and such that . Since the supports of and are mutually disjoint, we have
- 3.The values on γ 3. For u ∈ γ 3 we have (similarly as in the first case)
From Proposition 3, (4) and (5) we immediately get
3. Radial eigenfunctions
Note that the functions v and 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.
In particular, nontrivial Fučík eigenvalues on the first curve of the Fučík spectrum are not radial.
a contradiction which proves that .
a contradiction which proves that .
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 and are autonomous. The application of the Sturm Comparison Theorem is then more straightforward.
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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