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

Authors: Pavel Drábek
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We discuss nonlinear homogeneous eigenvalue problems and the variational characterization of their eigenvalues. We focus on the Ljusternik-Schnirelmann method, present one possible alternative to this method  and compare it with the Courant-Fischer minimax principle in the linear case. At the end we present a special nonlinear eigenvalue problem possessing an eigenvalue which allows the variational characterization but is not of Ljusternik-Schnirelmann type.


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## Body

The question of characterization of the set of all eigenvalues of nonlinear homogeneous eigenvalue problem is a very important problem of the calculus of variations that has been open since the 1970s, see, for example, [1]. While the well-known Courant-Fischer minimax principle provides complete characterization of the set of eigenvalues of the linear eigenvalue problem, its nonlinear counterparts, like the Ljusternik-Schnirelmann method, do not exhaust the entire set of eigenvalues in general. The purpose of this paper is to discuss this issue in the context of general homogeneous nonlinear eigenvalue problem (1). In particular, we want to explain that “variational characterization” is not a synonym for “Ljusternik-Schnirelmann characterization.” Indeed, this obfuscation is quite common in the literature (cf. [2]) and might lead to many misunderstandings.We present avariational argument in order to find an eigenvalue that is not characterized by the Ljusternik-Schnirelmann formula. For this purpose we present the periodic problem for the one-dimensional p-Laplacian from [3] as a concrete example and heuristically enlighten technically involved estimates. We show the local character of our variational characterization contrary to the global character of the Ljusternik-Schnirelmann method. This illustrates the striking difference between the linear and nonlinear problems. Note that also this “local procedure” does not yield all the eigenvalues of the problem in question.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.LetJ,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(1)J(u)-λS(u)=0
are called an eigenvalue and an eigenvector of the eigenvalue problem (1). Set(2)F(u)∶=〈J(u),u〉,G(u)∶=〈S(u),u〉,
where 〈·,·〉 stands for the duality pairing between X* and X. Assume that J and S are such that F and G are C1-functionals from X into ℝ,(3)〈F′(u),v〉=〈J(u),v〉,〈G′(u),v〉=〈S(u),v〉,S∶={u∈X:G(u)=1}
is a C1-manifold without boundary that does not contain the zero element 0∈X. The Lagrange multiplier method implies that λ0 is an eigenvalue of (1) and u0 is the corresponding eigenvector if and only if u0 is a critical point of F|𝒮 and F(u0)=λ0 (i.e., λ0 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.Set𝒜∶={A⊂X:Aisclosedandsymmetric}. The Krasnoselskii genus of a set A,  γ(A), is defined as follows:
(4)γ(A)∶={inf⁡⁡{m∈N:∃h∈C0(A,Rm∖{0}),hisodd},∞if⁡{m∈N:∃h∈C0(A,Rm∖{0}),hisodd}=∅.
For any k∈ℕ consider a family of sets defined by(5)Fk∶={A∈A:A⊂S,γ(A)≥k}.
Assume that γ(𝒮)=∞. Then, ℱk≠∅ for any k∈ℕ and it is invariant under any continuous and odd map from 𝒮 into itself.Let us assume thatF satisfies the Palais-Smale condition on 𝒮. The Ljusternik-Schnirelmann methodyields a sequence {λk}k=1∞⊂ℝ given by(6)λk∶=inf⁡A∈Fk⁡sup⁡u∈A⁡F(u).
Every λk is a critical levelof F subject to 𝒮, and it is achieved at some uk∈𝒮, that is,(7)(F|S)′(uk)=0
(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 bethe 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 theeigenvalue problem for the Dirichletp-Laplacian. Indeed, set X=W01,p(Ω), p>1, and(8)F(u)=∫Ω|∇u|pdx,G(u)=∫Ω|u|pdx,
where Ω⊂ℝN is a bounded domain. Then,(9)S={u∈X:‖u‖Lp(Ω)=1},
and a critical point u0∈𝒮,F(u0)=λ0 satisfies(10)〈F′(u0),v〉=λ0〈G′(u0),v〉
for any v∈X. But this is a weak formulation of the problem(11)-Δpu0-λ0|u0|p-2u0=0inΩ,u0=0on⁡∂Ω,
and, therefore, λ0 is an eigenvalue and u0 is the corresponding eigenvector (eigenfunction) of the Dirichlet p-Laplacian (11).Ifp≠2 and Ω⊂ℝN, N≥2, it is a long-standing open problemwhether or not the sequence{λk}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⊂ℝN, N≥1, and p=2 (i.e., the problem (11) is linear), the sequence {λn}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 ℱk can be replaced by an other family ℱ̃k, which is invariant with respect to any continuous odd semiflowΦ:𝒮×[0,∞)→𝒮 such that Φ(·,0)=id𝒮, Φ(·,t):𝒮→𝒮 is a homeomorphism for any fixed t≥0 and F(Φ(u,·)) is nonincreasing for any fixed u∈𝒮. Then,(12)λ̃k∶=inf⁡A∈F̃k⁡sup⁡u∈A⁡F(u),k∈N,
is a critical level of F|𝒮.We give aconcrete exampleof ℱ̃k below (cf. [6]). Let Sk-1 be the unit sphere in ℝk.Set(13)F̃k∶={A⊂S:∃h:Sk-1⟶S,A=h(Sk-1),hisoddandcontinuous}.
It is clear that ℱ̃k⊊ℱk, which implies(14)λk≤λ̃k,k∈N.It is also obvious thatλ1=λ̃1. We claim that λ2=λ̃2 as well. Indeed, for an A∈ℱ2 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(S1) for some continuous and odd map h of S1 into 𝒮. Since the neighborhood 𝒩(A) can be chosen small enough, the value of(15)max⁡u∈φ∪(-φ)⁡F(u)-sup⁡u∈A⁡F(u)
can be made smaller than any positive number. This implies λ̃2≤λ2, which proves the claim.The author is not aware of any assumptions laid onJ and S that would guarantee that also λk=λ̃k for k≥3 (cf. [6]).It is instructive to compare formula (12) with the classical Courant-Fisher minimax principle:(16)ck∶=inf⁡X′⊂Xdim⁡X′=k⁡max⁡u∈S∩X′⁡F(u)
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).LetX′⊂X,  dimX′=k. Then, A=𝒮∩X′ is homeomorphic to Sk-1, that is, A∈ℱ̃k and hence λ̃k≤ck. In order to prove the opposite inequality we need the following assertion.Lemma 1 (cf. [5]).
Any compact and symmetric subsetA of a Hilbert space, such that γ(A)=k contains at least k mutually orthogonal vectors u1,…,uk.Sinceγ(A)=kfor A∈ℱ̃k, there exist u1,…,uk∈A such that (Jui,uj)X=0 for i≠j. Set X′=Lin{u1,…,uk}. It follows from the linearity of J that max𝒮∩X′F(u)≤maxAF(u), which implies ck≤λ̃k. Together with the opposite inequality proved above we thus get(17)λ̃k=ck.
Similarly, one proves that also λk=ck.In particular, it follows from our discussion that the second Ljusternik-Schnirelmann eigenvalue of (1) can be characterized as follows:(18)λ2=inf⁡γ∈Csym⁡max⁡u∈γ⁡F(u),
where 𝒞sym 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 followingheuristics.SetΦ(γ)∶=maxu∈γF(u). Then, λ2=infγ∈𝒞symΦ(γ), that is, λ2 is a “global minimum” of Φ over the set 𝒞sym (cf. [5]). The idea is to find a “local minimum” of Φ over 𝒞sym, say μ2, which satisfies λ2<μ2<λ3, that is, μ2∉{λk}k=1∞. For this purpose we find a subset of curves 𝒞⊂𝒞sym, which is invariant with respect to any continuous odd semiflow and which satisfies(19)λ2=inf⁡γ∈Csym⁡max⁡u∈γ⁡F(u)<inf⁡γ∈C⁡max⁡u∈γ⁡F(u)=μ2<λ3.
The invariance of 𝒞 then guarantees that μ2 is a critical point of F|𝒮.Below we present aconcrete example. The technical details and precise estimates can be found in [3]. We set X=Wper1,p(-πp,πp), the Sobolev space of all (2πp)-periodic functions f:ℝ→ℝ such that the restriction of f to the interval [-πp,πp] belongs to W1,p(-πp,πp) and f(-πp)=f(πp). This space is endowed with the norm(20)∥f∥∶=(‖f′‖Lp(-πp,πp)p+‖f‖Lp(-πp,πp)p)1/p<∞.
Here,(21)πp∶=2∫01ds(1-sp)1/p=2πpsin⁡(π/p).
We also set(22)S∶={f∈X:∥f∥Lp(-πp,πp)=1}.
Let ɛ>0 be a fixed number, and let q:ℝ→ℝ be a given continuous (2πp)-periodic function of x∈ℝ.We defineJɛ,S:X→X* by
(23)〈Jɛ(u),v〉∶=∫-πpπp|u′|p-2u′v′dx+ɛ∫-πpπpq|u|p-2uvdx,〈S(u),v〉∶=∫-πpπp|u|p-2uvdx
for u,v∈X. We also set(24)Fɛ(u)∶=∫-πpπp|u′|pdx+ɛ∫-πpπpq|u|pdx.
The operator equation(25)Jɛ(u)-λS(u)=0
corresponds to the weak formulation of the following eigenvalue problem:
(26)-(|u′|p-2u′)′+ɛq(x)|u|p-2u=λ|u|p-2uinR,uis(2πp)-periodicinR.
For ɛ=0 the set of all eigenvalues of (26) is given by the sequence(27)λ0=0,λ2n-1=λ2n=(p-1)np,n=1,2,…,
and it is proved in [2] that each eigenvalue λk(k=0,1,2,…) has a Ljusternik-Schnirelmann variational characterization:(28)λk=inf⁡A∈Fk+1⁡max⁡u∈A⁡∫-πpπp|u′(x)|pdx.
The eigenvalue λ0 is simple, and the corresponding eigenfunction is a constant (and thus does not change the sign in [-πp,πp]). For any n=1,2,…, we have that λ2n-1=λ2n is not a simple eigenvalue. Given any fixed t∈ℝ, every function sinp(nx+t) of x∈ℝ is an eigenfunction associated with λ2n-1=λ2n. Here, sinpx for x∈[0,πp/2] is defined implicitly by the formula(29)x=∫0sin⁡px(1-sp)-1/pds
and then extended to an odd (about 0), (2πp)-periodic function on ℝ.Remark 2.
The reader should notice that only in the linear casep=2 do the corresponding eigenfunctions associated with λ2n-1=λ2n 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 λ2n-1=λ2n is a two-dimensional manifold, the linear hull of which is infinite-dimensional (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 Lp(-πp,πp), which is diffeomorphic to a circle, the linear hull of which is again infinite-dimensional.Forɛ>0,q≠0 the problem (26) possesses a sequence of eigenvalues tending to infinity and given by(30)λkɛ=inf⁡A∈Fk+1⁡max⁡u∈A⁡Fɛ(u)
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(31)λ2n-1ɛ≤λ2nɛ<λ2n+1ɛ≤λ2n+2ɛ,
where equalities may possibly occur. Eigenfunctions associated with λ2n-1ɛ and λ2nɛ have precisely 2n zeros in (-πp,πp], and, for each k=0,1,2,…, we have(32)lim⁡ɛ→0+⁡λkɛ=λk
(see [2]). It is also proved in [2] that for any integers m,n≥1 and for any ɛ>0, small enough, there exists a special(2πp)-periodic function q∈C1(ℝ) such that λ2n-1ɛ<λ2nɛ and the open interval (λ2n-1ɛ,λ2nɛ) 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 (λ2n-1,λ2n) 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 (λ1ɛ,λ2ɛ), provided ɛ is small enough.The first critical level of Ljusternik-Schnirelmann type (L.S. type for short) forF0 on 𝒮, 0=λ0=minu∈𝒮F0(u) is attained at ±φ0(φ0≡constant>0). The second critical level of L.S. type, λ1=p-1, is defined by the minimax formula(33)λ1=min⁡γ∈C1⁡max⁡τ∈[-1,1]⁡F0(γ(τ)),
where 𝒞1 is the set of all continuous curves in 𝒮 that connect the “south pole”-φ0 with the “north pole” φ0. The level λ1 is attained at the point u=γt(0) on the curve γt∈𝒞1(t∈ℝ) defined by(34)γt(τ)≔‖ht(τ)‖Lperpht(τ),τ∈[-1,1],
where(35)ht(τ)=τφ0+(1-τ2)e(t)(x),x∈R,
and e(t)(x)∶=(2πp/p)-1/psinp(x+t) is a normalized eigenfunction associated with λ1=λ2 (see [3]). Notice that the curves γt correspond to the “meridians” on 𝒮 with the points of  “longitude” t and the “latitude” τ.For construction of a special functionq=q(x) the following observation plays the key role: the function sinp:x→sinpx, from ℝ to ℝ, is real analytic in the open set ℝ∖{(1/2)kπp:k∈ℤ} and, furthermore, it is not C3 at the point  kπp (k∈ℤ) if 1<p<2 and it is not C3 at the point (k+1/2)πp if (k∈ℤ) 2<p<∞. Due to these facts a function q can be found that controls the “splitting process” that involves the eigenvalues λ1ɛ and λ2ɛ. In particular, q is constructed in such a way that the function of the independent variable t,(36)F(t;q)=∫-πpπpq(x)|sin⁡p⁡(x+t)|pdx,
has a global minimizer tmin∈[0,πp] and a local minimizer t̅min∈(0,πp) such that(37)F(tmin⁡;q)<F(t¯min⁡;q).
This property together with fine estimates for p>2 (see [3]) allows us to prove the local hyperbolic geometry of the functional Fɛ on 𝒮. Namely, thanks to (37), we can "localize" the set of curves 𝒞2⊊𝒞1 that pass through a suitable open neighborhood of the function e(t¯min) in the topology of Lp(-πp,πp). The “inf” in the variational characterization of λ1ɛ is then approached “away from” curves that belong to 𝒞2. The family of curves 𝒞2 is then proved to be invariant with respect to continuous odd semiflow on 𝒮 and(38)λ1ɛ=inf⁡γ∈C1⁡max⁡τ∈[-1,1]⁡Fɛ(γ(τ))<inf⁡γ∈C2⁡max⁡τ∈[-1,1]⁡Fɛ(γ(τ))<λ2ɛ
holds for ɛ>0 small enough. Then,(39)λ1ɛ<λ̃1ɛ=inf⁡γ∈C2⁡max⁡τ∈[-1,1]⁡Fɛ(γ(τ))<λ2ɛ
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 sinx. 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.



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