On the Variational Eigenvalues Which Are Not of Ljusternik-Schnirelmann Type

and Applied Analysis 3 Every λk is a critical level of F subject to S, and it is achieved at some uk ∈ S, that is, F|S ′ uk 0 7 cf. 4, 5 . It follows that λk is an eigenvalue of 1 and uk is the corresponding eigenvector. In general, the sequence {λk}k 1 given by 6 does not exhaust the set of all critical levels of F|S, and thus it might not be the set of all eigenvalues of 1 . An eigenvalue of 1 that allows the characterization 6 is called an eigenvalue of Ljusternik-Schnirelmann type. The model example of the abstract setting presented above is the eigenvalue problem for the Dirichlet p-Laplacian. Indeed, set X W 0 Ω , p > 1, and


Abstract and Applied Analysis
The novelty of this paper is thus to point out some connections might be hidden in more technically complicated papers and to remind some challenging problems of the calculus of variations. We also rigorously show that the second eigenvalue of Ljusternik-Schnirelmann type for 1 can be found as a minimax over a special class of curves rather than the family of sets of genus greater than or equal to 2. This fact makes the construction of the second eigenvalue more geometrically transparent and might be useful in numerical approach to find its concrete value.
Let J, S : X → X * be odd and p − 1 -homogeneous operator from a Banach space X into its dual, p > 1. A real number λ and an element u ∈ X, u / 0, for which where ·, · stands for the duality pairing between X * and X. Assume that J and S are such that F and G are C 1 -functionals from X into R, is a C 1 -manifold without boundary that does not contain the zero element 0 ∈ X. The Lagrange multiplier method implies that λ 0 is an eigenvalue of 1 and u 0 is the corresponding eigenvector if and only if u 0 is a critical point of F| S and F u 0 λ 0 i.e., λ 0 is the corresponding critical level . There are several methods to find critical points of F on S. However, it seems that with the exception of some special situations e.g., J and S are linear operators none of the "nonlinear" methods describes the entire set of critical levels and critical points of F| S . One of the most frequently used methods is based on the notion of the Krasnoselskii genus.
Set A : {A ⊂ X : A is closed and symmetric}. The Krasnoselskii genus of a set A, γ A , is defined as follows: For any k ∈ N consider a family of sets defined by Assume that γ S ∞. Then, F k / ∅ for any k ∈ N and it is invariant under any continuous and odd map from S into itself.
Let us assume that F satisfies the Palais-Smale condition on S. The Ljusternik-Schnirelmann method yields a sequence {λ k } ∞ k 1 ⊂ R given by Abstract and Applied Analysis 3 Every λ k is a critical level of F subject to S, and it is achieved at some u k ∈ S, that is, cf. 4, 5 . It follows that λ k is an eigenvalue of 1 and u k is the corresponding eigenvector.
In general, the sequence {λ k } ∞ k 1 given by 6 does not exhaust the set of all critical levels of F| S , and thus it might not be the set of all eigenvalues of 1 . An eigenvalue of 1 that allows the characterization 6 is called an eigenvalue of Ljusternik-Schnirelmann type.
The model example of the abstract setting presented above is the eigenvalue problem for the Dirichlet p-Laplacian. Indeed, set X W 1,p 0 Ω , p > 1, and and a critical point u 0 ∈ S, F u 0 λ 0 satisfies for any v ∈ X. But this is a weak formulation of the problem and, therefore, λ 0 is an eigenvalue and u 0 is the corresponding eigenvector eigenfunction of the Dirichlet p-Laplacian 11 . If p / 2 and Ω ⊂ R N , N ≥ 2, it is a long-standing open problem whether or not the sequence{λ k } ∞ k 1 given by 6 exhausts the entire set of eigenvalues of 11 . On the other hand, if Ω 0, 1 , p > 1, or Ω ⊂ R N , N ≥ 1, and p 2 i.e., the problem 11 is linear , the sequence {λ n } ∞ n 1 given by 6 is the set of all eigenvalues of 11 . To prove this fact in the former case the ODE techniques uniqueness of the solution of an associated initial value problem are employed while in the latter case the linearity of the problem plays the key role.
There are several other variational characterizations similar to 6 . Namely, the family of sets F k can be replaced by an other family F k , which is invariant with respect to any continuous odd semiflow Φ : S × 0, ∞ → S such that Φ ·, 0 id S , Φ ·, t : S → S is a homeomorphism for any fixed t ≥ 0 and F Φ u, · is nonincreasing for any fixed u ∈ S.

Abstract and Applied Analysis
Then, is a critical level of F| S . We give a concrete example of F k below cf. 6 . Let S k−1 be the unit sphere in R k . Set It is clear that F k F k , which implies It is also obvious that λ 1 λ 1 . We claim that λ 2 λ 2 as well. Indeed, for an A ∈ F 2 there exists arbitrarily small symmetric neighborhood N A ⊂ S such that γ N A ≥ 2 see, 7, Proposition 2.3 . Without loss of generality we may assume that N A is a pathwise connected set in S otherwise γ N A 1, a contradiction . Choosing u ∈ N A , we have −u ∈ N A by symmetry, and there is a continuous path ϕ ∈ N A from u to −u. By symmetry of N A we also have −ϕ ⊂ N A . But ϕ ∪ −ϕ h S 1 for some continuous and odd map h of S 1 into S. Since the neighborhood N A can be chosen small enough, the value of can be made smaller than any positive number. This implies λ 2 ≤ λ 2 , which proves the claim. The author is not aware of any assumptions laid on J and S that would guarantee that also λ k λ k for k ≥ 3 cf. 6 .
It is instructive to compare formula 12 with the classical Courant-Fisher minimax principle: under the assumptions that J, S are linear operators, X is a Hilbert space and Ju, u X defines an equivalent inner product on X cf. problem 11 with p 2 . Let X ⊂ X, dim X k. Then, A S ∩ X is homeomorphic to S k−1 , that is, A ∈ F k and hence λ k ≤ c k . In order to prove the opposite inequality we need the following assertion. Lemma 1 cf. 5 . Any compact and symmetric subset A of a Hilbert space, such that γ A k contains at least k mutually orthogonal vectors u 1 , . . . , u k .
Since γ A k for A ∈ F k , there exist u 1 , . . . , u k ∈ A such that Ju i , u j X 0 for i / j. Set X Lin{u 1 , . . . , u k }. It follows from the linearity of J that max S∩X F u ≤ max A F u , which implies c k ≤ λ k . Together with the opposite inequality proved above we thus get Similarly, one proves that also λ k c k . In particular, it follows from our discussion that the second Ljusternik-Schnirelmann eigenvalue of 1 can be characterized as follows: where C sym is the family of all continuous symmetric and closed curves in S. The idea of construction of an eigenvalue that is not of Ljusternik-Schnirelmann type comes from the following heuristics.
Set Φ γ : max u∈γ F u . Then, λ 2 inf γ∈C sym Φ γ , that is, λ 2 is a "global minimum" of Φ over the set C sym cf. 5 . The idea is to find a "local minimum" of Φ over C sym , say μ 2 , which satisfies λ 2 < μ 2 < λ 3 , that is, μ 2 / ∈ {λ k } ∞ k 1 . For this purpose we find a subset of curves C ⊂ C sym , which is invariant with respect to any continuous odd semiflow and which satisfies The invariance of C then guarantees that μ 2 is a critical point of F| S . Below we present a concrete example. The technical details and precise estimates can be found in 3 . We set X W 1,p per −π p , π p , the Sobolev space of all 2π p -periodic functions f : R → R such that the restriction of f to the interval −π p , π p belongs to W 1,p −π p , π p and f −π p f π p . This space is endowed with the norm Here, We also set S : f ∈ X : f L p −π p ,π p 1 . 22 Let ε > 0 be a fixed number, and let q : R → R be a given continuous 2π p -periodic function of x ∈ R.

Abstract and Applied Analysis
We define J ε , S : X → X * by J ε u , v : for u, v ∈ X. We also set F ε u : The operator equation corresponds to the weak formulation of the following eigenvalue problem:

26
For ε 0 the set of all eigenvalues of 26 is given by the sequence λ 0 0, λ 2n−1 λ 2n p − 1 n p , n 1, 2, . . . , 27 and it is proved in 2 that each eigenvalue λ k k 0, 1, 2, . . . has a Ljusternik-Schnirelmann variational characterization: The eigenvalue λ 0 is simple, and the corresponding eigenfunction is a constant and thus does not change the sign in −π p , π p . For any n 1, 2, . . ., we have that λ 2n−1 λ 2n is not a simple eigenvalue. Given any fixed t ∈ R, every function sin p nx t of x ∈ R is an eigenfunction associated with λ 2n−1 λ 2n . Here, sin p x for x ∈ 0, π p /2 is defined implicitly by the formula Remark 2. The reader should notice that only in the linear case p 2 do the corresponding eigenfunctions associated with λ 2n−1 λ 2n form a linear space of dimension two. In contrast, if p / 2, then the multiplicity should be understood in the sense of the Ljusternik-Schnirelmann variational characterization 28 . In fact, in this case the set of all eigenfunctions associated with each eigenvalue λ 2n−1 λ 2n is a two-dimensional manifold, the linear hull of which is infinitedimensional cf. 2 . If these eigenfunctions are normalized by a scalar multiplication factor in order to belong to S, they form a one-dimensional submanifold in L p −π p , π p , which is diffeomorphic to a circle, the linear hull of which is again infinite-dimensional.
For ε > 0, q / 0 the problem 26 possesses a sequence of eigenvalues tending to infinity and given by for k 0, 1, 2, . . .. The first eigenvalue is simple, and the corresponding eigenfunction does not change sign. For each n 1, 2, 3, . . . we have where equalities may possibly occur. Eigenfunctions associated with λ ε 2n−1 and λ ε 2n have precisely 2n zeros in −π p , π p , and, for each k 0, 1, 2, . . ., we have It is also proved in 2 that for any integers m, n ≥ 1 and for any ε > 0, small enough, there exists a special 2π p -periodic function q ∈ C 1 R such that λ ε 2n−1 < λ ε 2n and the open interval λ ε 2n−1 , λ ε 2n contains at least m eigenvalues of 26 . In particular, these eigenvalues do not allow the variational characterization 30 . In the paper 3 we have shown that in contrast with the eigenvalues of Ljusternik-Schnirelmann type that are "globally variational," some of the above mentioned "new" eigenvalues from λ 2n−1 , λ 2n have a variational characterization that has a "local character," meaning that the minimum part inf of the minimax formula 30 is taken only locally in the sense mentioned above. In order to illustrate our method we focus on n 1, that is, we find "local variational" eigenvalue of 26 in the interval λ ε 1 , λ ε 2 , provided ε is small enough. The first critical level of Ljusternik-Schnirelmann type L.S. type for short for F 0 on S, 0 λ 0 min u∈S F 0 u is attained at ±ϕ 0 ϕ 0 ≡ constant > 0 . The second critical level of L.S. type, λ 1 p − 1, is defined by the minimax formula where C 1 is the set of all continuous curves in S that connect the "south pole"-ϕ 0 with the "north pole" ϕ 0 . The level λ 1 is attained at the point u γ t 0 on the curve γ t ∈ C 1 t ∈ R defined by and e t x : 2π p /p −1/p sin p x t is a normalized eigenfunction associated with λ 1 λ 2 see 3 . Notice that the curves γ t correspond to the "meridians" on S with the points of "longitude" t and the "latitude" τ. For construction of a special function q q x the following observation plays the key role: the function sin p : x → sin p x, from R to R, is real analytic in the open set R \ { 1/2 kπ p : k ∈ Z} and, furthermore, it is not C 3 at the point kπ p k ∈ Z if 1 < p < 2 and it is not C 3 at the point k 1/2 π p if k ∈ Z 2 < p < ∞. Due to these facts a function q can be found that controls the "splitting process" that involves the eigenvalues λ ε 1 and λ ε 2 . In particular, q is constructed in such a way that the function of the independent variable t, has a global minimizer t min ∈ 0, π p and a local minimizer t min ∈ 0, π p such that F t min ; q < F t min ; q . 37 This property together with fine estimates for p > 2 see 3 allows us to prove the local hyperbolic geometry of the functional F ε on S. Namely, thanks to 37 , we can "localize" the set of curves C 2 C 1 that pass through a suitable open neighborhood of the function e t min in the topology of L p −π p , π p . The "inf" in the variational characterization of λ ε 1 is then approached "away from" curves that belong to C 2 . The family of curves C 2 is then proved to be invariant with respect to continuous odd semiflow on S and is the desired variational eigenvalue that is not of the L.S. type. This construction is not possible in the linear case, p 2. Besides the fact that all eigenvalues of the linear problem are of L.S. type, this follows from the analyticity of the function sin x. For this reason the functional F with desired properties does not exist in the case p 2, which illustrates the striking difference between the linear and nonlinear eigenvalue problem.