Abstract
It is wellknown that the second eigenvalue λ_{2 }of the Dirichlet Laplacian on the ball is not radial. Recently, Bartsch, Weth and Willem proved that the same conclusion holds true for the socalled nontrivial (sign changing) Fučík eigenvalues on the first curve of the Fučík spectrum which are close to the point (λ_{2}, λ_{2}). 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 eigenfunctions1. Introduction
Let Ω ⊂ ℝ^{N }be a bounded domain, N ≥ 2. The Fučík spectrum of Δ on
has a nontrivial solution
we have that λ_{1 }< η(λ) < ∞ for every λ (>λ_{1}), and the curve
consists of nontrivial Fučík eigenvalues. Moreover, it was proved in [2] that
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
The original purpose of this paper was to prove that the above conjecture holds true for all
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
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.
Figure 1. The first two Fučík curves.
We show that the point of intersection of this line and
Then
By the Lagrange multipliers rule,
for all
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
From now on we assume s ≥ 0, which is no restriction since Σ is clearly symmetric with respect to the diagonal.
The first eigenvalue λ_{1 }of Δ on
It is well known that λ_{1 }> 0, simple and admits an eigenfunction
and
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
(i) φ_{1 }is a strict global minimum of
(ii) φ_{1 }is a strict local minimum of
(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 λ_{1}(Ω) on the domain Ω. The following proposition follows immediately from the variational characterization of λ_{1 }given by (3) and the properties of the corresponding eigenfunction φ_{1}.
Proposition 1. λ_{1}(Ω_{2}) < λ_{1}(Ω_{1}) whenever Ω_{i}, i = 1, 2, are bounded domains satisfying Ω_{1 }⊆ Ω_{2 }and meas(Ω_{1}) < meas(Ω_{2}).
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
Figure 2. The ball decomposition
Moreover, from the variational characterization (3), the following properties of the function
follow immediately.
Proposition 2. The function (6) is continuous and strictly decreasing on (0, 1), it maps (0, 1) onto (λ_{1}(B), ∞) and
In particular, it follows from Proposition 2 that, given s ≥ 0, there exists a unique
Let
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
Proposition 3.
For the proof we need socalled raystrict convexity of the functional
defined on
We say that
where the equality holds if and only if v_{1 }and v_{2 }are colinear.
Lemma 4 (see [5, p. 132]). The functional
Proof of Proposition 3.
1. The values on γ_{1}. For u ∈ γ_{1 }we have
by Lemma 4 (with Ω := B), (3) and (7).
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
by (7).
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
Proposition 5. Given s ≥ 0, we have
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
Set r = x and write v(r) = u(x). It follows from the regularity theory that (10) is equivalent to the singular problem
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):
The function v is a solution of (12) if and only if
Note that the functions v and
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 (λ_{1}, λ_{1 } s) and (λ_{1 }+ s, λ_{1}). 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 u^{1 }and the latter one by u^{2}, respectively. Let (λ^{1 }+ s, λ^{1}) and (λ^{2 }+ s, λ^{2}) be Fučík eigenvalues associated with u^{1 }and u^{2}, 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
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 d^{1 }∈ (0, 1) such that v^{1}(r) is a solution of
and
After the substitution
and
Let u_{1 }= u_{1}(x) and u_{2 }= u_{2}(x) be the principal positive eigenfunctions associated with
and
After the substitution
and
The substitution
Let us assume that
It follows that (18) is a Sturm majorant for (17) on the interval
Since we also have
a contradiction which proves that
Similarly as above, there exists d^{2 }∈ (0, 1) such that v^{2 }is a solution of
and
After the substitution
and
Assume that
As above we obtain
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
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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