Yesterday, during a discussion with P. Drabek, I have learned about one nice fact (see Lemma below) on the multiplicity of eigenvalues for the following one-dimensional (nonlinear) eigenvalue problem for the weighted $p$-Laplacian:

\begin{equation}\label{1} \begin{cases} -(a(x)|u’|^{p-2} u’)’ = \lambda b(x) |u|^{p-2}u \quad {\rm in}\ (0,1), \\ u(0) = u(1) = 0, \end{cases} \end{equation}

where $p \geq 2$, $a,b \in C^1(a,b)$, $a,b > 0$ on $(0,1)$. Denote by $\{\lambda_k\}$ a set of eigenvalues of \eqref{1}. (In fact, $\{\lambda_k\}$ is countable, see below.)

Unlike the linear case $p=2$, in the general nonlinear case $p \neq 2$ it is necessary to specify what one means under the multiplicity of an eigenvalue $\lambda_k$. Naturally, there are two types of the multiplicity:

  • Algebraic multiplicity, say $m$, means that there exist $m$ eigenvalues $\lambda_{l}, \dots, \lambda_{l+m-1}$ with $l \leq k \leq l+m-1$ such that
  • Geometric multiplicity, say $m$, means that there exist $m$ distinct eigenfunctions associated with $\lambda_k$. In fact, it is not entirely clear what it should be meant under “distinct”. Definitely, each eigenfunction has to be defined uniquely up to scaling. Moreover, in the linear case $p=2$, the eigenfunctions have to be linearly independent. However, in the case $p \neq 2$, it seems to be unnecessary to demand the linear independency from eigenfuctions. That is, if $u_1$ and $u_2$ are linearly independent and both associated with $\lambda_k$, and, by some chance, $c_1 u_1 + c_2 u_2$ is again an eigenfunction associated with $\lambda_k$ for some $c_1,c_2 \neq 0$, then all three eigenfunctions $u_1$, $u_2$, $c_1 u_1 + c_2 u_2$ can be considered as distinct.

Just by definition, the algebraic multiplicity implies the geometric one. The natural (and seems to be open) question is the converse assertion: Does the geometric multiplicity imply the algebraic one? Surely, this question is worth for a higher-dimensional problem, too.

In this direction, the following interesting result was proved by J. Nečas (see Appendix V, Theorme 1.1, p. 233 in [1]):

Lemma. All eingevalues of \eqref{1} form a countable set $0 < \lambda_1 < \lambda_2 < \ldots$ such that $\lim\limits_{k\to\infty} \lambda_k = \infty$. All (normed) eigenfunctions are isolated. Moreover, each $\lambda_k$ has only a finite number of (normed) eigenfunctions.


This lemma asserts that any eigenvalue of \eqref{1} has the algebraic multiplicity exactly one, and the geometric multiplicity at least one. It is clear that if $p=2$ or the weights $a(x),b(x)$ are constants, then the geometric multiplicty is exactly one, too. However, the proof of Lemma goes by contradiction, and the authors of [1] do not provide any example of coefficients such that a nontrivial geometric multiplicty occurs. (It is still possible that the geometric multiplicity equals one all the time.)


There are two more remarks on this topic. Consider the problem

where $Q$ is a 2D square. In our article [2], we stated the following question. A numerical evidence [3] supports the conjecture that $\lambda_2 < \lambda_3$ if $p\neq 2$, unlike the linear case where equality trivially holds. On the other hand, if the nodal set of a second eigenfunction is a middle line or a diagonal of the square, then the rotation of this eigenfunction by angle $\frac{\pi}{2}$ is also a second eigenfunction. These two second eigenfunctions are geometrically distinct, but could correspond to the eigenvalue of the algebraic multiplicity one. However, there is no rigorous proof so far.


A rigorous example where the geometric multiplicity is larger than the algebraic one occurs if we consider the following superlinear problem

where $2<q<2^*$ and $A$ is a planar (concentric) annuli. It was observed by Coffman in [4] that if $A$ is thin enough, then a minimizer $v$ of

is nonradial and positive. (Moreover $v$ is a least-energy solution of the problem, which is an analog of the first eigenfunction.) Thus, any $SO(2)$ rotation of $v$ is again a minimizer of $\mu_1$, and hence a least-energy solution of the problem. That is, if the word “distinct” in our definition of the geometric multiplicity does include rotations of the same eigenfunction, then we have infinite geometric multiplicity of $\mu_1$. At the same time, the algebraic multiplicity of $\mu_1$ is finite if we consider $\mu_1$ as an element of a sequence of Lusternik-Schnirelmann critical points of the Rayleigh quotient in $\mu_1$. (Yes, it is a kind of cheating, but there is no complete description of the spectrum to such problems, so…)

(See also [5] and [6] and references therein for the development of multiplicity and nonradialiry results for nonlinear case $p>1$ and higher dimension $N \geq 2$).


Bibliography

  1. Fučik, S., Nečas, J., Souček, J., & Souček, V. (1973). Spectral analysis of nonlinear operators. Springer.  2

  2. Audoux, B., Bobkov, V., & Parini, E. (2018). On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian. Topological Methods in Nonlinear Analysis. 

  3. X. Yao, J. Zhou, Numerical methods for computing nonlinear eigenpairs. I. Iso-homogeneous cases, SIAM J. Sci. Comput. 29 (2007), no. 4, 1355–1374. 

  4. Coffman, C. V. (1984). A non-linear boundary value problem with many positive solutions. Journal of differential equations, 54(3), 429-437. 

  5. Nazarov, A. I. (2004). On Solutions to the Dirichlet problem for an equation with $p$-Laplacian in a spherical layer. In Proc. St.-Petersburg Math. Soc (Vol. 10, pp. 33-62). 

  6. Kolonitskii, S. (2011). Multiplicity of solutions of the Dirichlet problem for an equation with the $p$-Laplacian in a three-dimensional spherical layer. St. Petersburg Mathematical Journal, 22(3), 485-495.