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 is defined as a set Σ of those (λ_{+}, λ_{}) ∈ ℝ^{2 }such that the Dirichlet problem
has a nontrivial solution . In particular, if λ_{1 }< λ_{2 }< ⋯ 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 {λ_{1}} × ℝ and ℝ × {λ_{1}}. Following [1, p. 15], we call the elements of Σ \ ({λ_{1}} × ℝ ∪ ℝ × {λ_{1}}) nontrivial Fučík eigenvalues. It was proved in [2] that there exists a first curve of nontrivial Fučík eigenvalues in the sense that, defining η: (λ_{1}, ∞) → ℝ by
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 is a continuous and strictly decreasing curve which contains the point (λ_{2}, λ_{2}) 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 [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
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 corresponds to the critical value of some constrained functional (cf. [4, p. 214]). To this end we define the functional
Then is a C^{1}functional on and we look for the critical points of the restriction of to
By the Lagrange multipliers rule, is a critical point of if and only if there exists t ∈ ℝ such that
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 is defined as
It is well known that λ_{1 }> 0, simple and admits an eigenfunction with φ_{1 }satisfying φ_{1}(x) > 0 for x ∈ Ω. Let
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 are classified as follows.
(i) 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}).
(ii) 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}).
(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 such that
Let and be positive principle eigenvalues associated with and , respectively. We extend both functions on the entire B by setting on , on and then normalize them by , . Our aim is to construct a special curve γ ∈ Γ on which the values of stay below . Actually, the curve γ connects φ_{1 }with (φ_{1}) and passes through and . For this purpose we set γ = γ_{1 }∪ γ_{2 }∪ γ_{3}, where
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
For the proof we need socalled raystrict convexity of the functional
defined on
We say that is raystrictly convex if for all τ ∈ (0, 1) and v_{1}, v_{2 }∈ V_{+ }we have
where the equality holds if and only if v_{1 }and v_{2 }are colinear.
Lemma 4 (see [5, p. 132]). The functional defined by (8) is raystrictly convex.
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 . Since the supports of and are mutually disjoint, we have
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 is a solution of
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 (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 , is a solution of
and
Let u_{1 }= u_{1}(x) and u_{2 }= u_{2}(x) be the principal positive eigenfunctions associated with and , 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 and 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
and
After the substitution , i = 1, 2, we have
and
The substitution transforms (15) to
Let us assume that and that d^{1 }> d_{s}. Choose and set . Then solves
It follows that (18) is a Sturm majorant for (17) on the interval and on . Since and , , we have a contradiction with the Sturm Separation Theorem (see [7, Cor. 3.1, p. 335]). Hence d^{1 }≤ d_{s}. Similar application of the Strum Separation Theorem to (14) and (16) now yields
Since we also have , it follows from (7) and (19) that
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 , is a solution of
and
Assume that and that 1 d_{s }> d^{2}. Similar arguments based on the Sturm Comparison Theorem yield first that 1 d_{s }≤ d^{2 }(i.e., 1  d^{2 }≤ d_{s}), and then (16), (21) 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 and 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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