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Does the Lusternik-Schnirelmann eigenvalues exhaust the spectrum of the $p$-Laplacian?
Step-by-step mathematics solution: Does the Lusternik-Schnirelmann eigenvalues exhaust the spectrum of the $p$-Laplacian?
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1. What the question is asking (in plain language)
We have the Dirichlet (p)-Laplacian
[ -\Delta _p u:= -\operatorname{div}!\bigl(|\nabla u|^{p-2}\nabla u\bigr),\qquad 1<p<\infty , ]
on a bounded open set (\Omega\subset\mathbb{R}^{d}) with the boundary condition (u=0) on (\partial\Omega).
An eigenvalue (\lambda) is a real number for which there exists a non‑zero (u\in W^{1,p}_0(\Omega)) solving
[ -\Delta _p u = \lambda |u|^{p-2}u\quad\text{in }\Omega . ]
Using the Lusternik–Schnirelmann (LS) min–max principle one can construct an increasing sequence of numbers
[ 0<\lambda_{1}(p)\le \lambda_{2}(p)\le\cdots ,\qquad \lambda_{k}(p)\xrightarrow{k\to\infty}\infty , ]
called variational eigenvalues.
For the linear case (p=2) the spectral theorem tells us that every eigenvalue of the Laplacian appears in this list (counting multiplicities).
The question is:
When (p\neq 2), does the LS list ({\lambda_k(p)}_{k\ge1}) contain all eigenvalues of the (p)-Laplacian, or can there be “extra’’ eigenvalues that are not obtained by the LS min–max construction?
In other words, does the LS sequence exhaust the whole spectrum (the set of all eigenvalues) of the nonlinear operator (-\Delta _p)?
2. Detailed answer
2.1 Preliminaries
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Rayleigh quotient.
For (u\in W^{1,p}_0(\Omega)\setminus{0}) define[ \mathcal R(u)=\frac{\displaystyle\int_{\Omega}|\nabla u|^{p}\,dx} {\displaystyle\int_{\Omega}|u|^{p}\,dx}. ]
The functional (\mathcal R) is homogeneous of degree 0: (\mathcal R(tu)=\mathcal R(u)) for every (t\neq0).
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Unit sphere.
Let[ S:=\bigl{u\in W^{1,p}0(\Omega):\ |u|{L^{p}(\Omega)}=1\bigr}. ]
Then (\mathcal R) restricted to (S) is a (C^{1}) functional whose critical points are precisely the eigenfunctions of (-\Delta _p) (with eigenvalue equal to the critical value).
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Genus.
If (A\subset S) is closed, symmetric ((u\in A\Rightarrow -u\in A)), its Krasnosel’skiĭ genus (\gamma(A)) is the smallest integer (k) such that there exists an odd continuous map (A\to\mathbb S^{k-1}) (or (+\infty) if no such (k) exists). -
Variational eigenvalues (LS definition).
For each integer (k\ge 1) set[ \lambda_k(p):=\inf_{A\in\mathcal F_k}\ \sup_{u\in A}\mathcal R(u), \qquad \mathcal F_k:={A\subset S:\ A\ \text{closed, symmetric, }\gamma(A)\ge k}. ]
The LS theorem (or the “genus min–max principle’’) guarantees that each (\lambda_k(p)) is a critical value of (\mathcal R) on (S); hence (\lambda_k(p)) is an eigenvalue of (-\Delta _p).
The question is whether every eigenvalue (\lambda) of (-\Delta _p) can be written as (\lambda_k(p)) for some (k).
2.2 Main theorem (classical result)
Theorem (Lusternik–Schnirelmann exhaustivity for the Dirichlet (p)-Laplacian).
Let (\Omega\subset\mathbb R^{d}) be bounded and have a Lipschitz boundary. Then the set of eigenvalues of the Dirichlet problem
(-\Delta p u = \lambda |u|^{p-2}u) in (\Omega,\; u=0) on (\partial\Omega)
coincides with the LS sequence ({\lambda_k(p)}{k\ge1}).
In particular the spectrum is a countable, unbounded, increasing list (counting multiplicities).
The theorem is proved in several papers; a concise reference is
- A. Anane, “On the second eigenvalue of the (p)-Laplacian”, Nonlinear Anal. 16 (1991), 105–117,
- P. Drábek, “Solvability and spectrum of the (p)-Laplacian”, J. Funct. Anal. 131 (1995), 315–341,
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J. Lindqvist, **“On the equation (\Delta _p u + \lambda u ^{p-2}u=0)”*, *Proc. Amer. Math. Soc. 109 (1990), 157–164.
Below we outline the essential ideas of the proof.
2.3 Sketch of the proof
Step 1 – Palais–Smale condition for the Rayleigh quotient
Define
[ \Phi(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^{p}\,dx -\frac{\lambda}{p}\int_{\Omega}|u|^{p}\,dx . ]
Restricted to the sphere (S) the functional (\mathcal R) (or (\Phi)) satisfies the Palais–Smale condition: every sequence ({u_n}\subset S) with (\mathcal R(u_n)) bounded and (\mathcal R’(u_n)\to0) possesses a convergent subsequence in (W^{1,p}_0(\Omega)). The proof uses the compact embedding (W^{1,p}_0(\Omega)\hookrightarrow L^{p}(\Omega)) (Rellich‑Kondrachov) and the strict monotonicity of the map (z\mapsto |z|^{p-2}z).
This compactness is the heart of the LS theory for nonlinear operators: it guarantees that the min–max values (\lambda_k(p)) are indeed critical values.
Step 2 – Existence of a critical point at each (\lambda_k(p))
By definition, [ \lambda_k(p)=\inf_{A\in\mathcal F_k}\sup_{u\in A}\mathcal R(u). ] Take a minimizing sequence of sets (A_n\in\mathcal F_k) with (\sup_{u\in A_n}\mathcal R(u)\downarrow\lambda_k(p)). Using the deformation lemma (valid because of the Palais–Smale condition) one can construct a Palais–Smale sequence for (\mathcal R) at level (\lambda_k(p)). By Step 1 it converges to a critical point (u_k\neq0). Hence (\lambda_k(p)) is an eigenvalue.
Step 3 – Any eigenvalue appears in the LS list
Let (\lambda) be an eigenvalue and let [ E_\lambda:={u\in S:\ \mathcal R(u)=\lambda} ] be the set of normalized eigenfunctions belonging to (\lambda). (E_\lambda) is closed, symmetric, and compact (again because of the compact embedding). Moreover, the genus of (E_\lambda) is a finite integer that we denote by (\gamma(E_\lambda)).
Now consider the LS min–max values: [ \lambda_{k}(p)=\inf_{A\in\mathcal F_k}\sup_{u\in A}\mathcal R(u). ]
Because (E_\lambda) is a symmetric compact set, (\gamma(E_\lambda)=m) implies (E_\lambda\in\mathcal F_m) but (E_\lambda\notin\mathcal F_{m+1}). Consequently
[ \lambda_m(p)\le \lambda\le\lambda_{m+1}(p). ]
If the inequality were strict, i.e. (\lambda_m(p)<\lambda<\lambda_{m+1}(p)), the mountain‑pass type deformation (or a linking argument) would produce a new critical value strictly between (\lambda_m(p)) and (\lambda_{m+1}(p)), contradicting the definition of (\lambda_{m+1}(p)) as the least level that can be forced by a set of genus (\ge m+1). Therefore the only possibility is
[ \boxed{\ \lambda=\lambda_m(p)\ } . ]
Thus every eigenvalue coincides with one of the LS numbers.
Step 4 – Counting multiplicities
If an eigenvalue (\lambda) has multiplicity (r) (i.e. the linear span of its eigenfunctions has dimension (r)), then (\gamma(E_\lambda)=r). Consequently the same eigenvalue appears exactly (r) times in the list ({\lambda_k(p)}), just as in the linear case.
2.4 Consequences
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The spectrum of the Dirichlet (p)-Laplacian is discrete, consists of a countable set of real numbers, and has no accumulation point except (+\infty).
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The LS sequence ({\lambda_k(p)}) exhausts the whole spectrum; there are no hidden eigenvalues outside this variational list.
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The first eigenvalue (\lambda_1(p)) is simple, isolated and characterized by the Rayleigh quotient minimization. Higher eigenvalues can be multiple; their multiplicities are detected by the genus of the corresponding critical sets.
3. Final
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