Topological Methods in Nonlinear Analysis

Nonlinear eigenvalue problems admitting eigenfunctions with known geometric properties

Michael Heid and Hans-Peter Heinz

Full-text: Open access


We consider nonlinear eigenvalue problems of the form $$ A_0 y + B(y) y = \lambda y \tag{$*$} $$ in a real Hilbert space $\mathcal H$, where $A_0$ is a semi-bounded self-adjoint operator and, for every $y$ from a certain dense subspace $X$ of $\mathcal H$, $B(y)$ is a bounded symmetric linear operator. The left hand side is assumed to be the gradient of a functional $\psi \in C^1(x)$, and the associated linear problems $$ A_0 v + B(y) v = \mu v \tag{$**$} $$ are supposed to have discrete spectrum $(y \in X)$. We present a new topological method which permits, under appropriate assumptions, to construct solutions of ($*$) on a sphere $S_R := \{ y \in X \mid \|y\|_{\mathcal H} = R\}$ whose $\psi$-value is the $n$th Ljusternik-Schnirelman level of $\psi |_{S_R}$ and whose corresponding eigenvalue is the $n$-th eigenvalue of the associated linear problem ($**$), where $R > 0$ and $n \in \mathbb N$ are given. In applications, the eigenfunctions thus found share any geometric property enjoyed by an $n$-th eigenfunction of a linear problem of the form ($**$). We discuss applications to nonlinear Sturm-Liouville problems, to the nonlinear Hill's equation, to periodic solutions of second-order systems, and to elliptic partial differential equations with radial symmetry.

Article information

Topol. Methods Nonlinear Anal., Volume 13, Number 1 (1999), 17-51.

First available in Project Euclid: 29 September 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Heid, Michael; Heinz, Hans-Peter. Nonlinear eigenvalue problems admitting eigenfunctions with known geometric properties. Topol. Methods Nonlinear Anal. 13 (1999), no. 1, 17--51.

Export citation


  • M. Atiyah, K-Theory, W. A. Benjamin, New York–Amsterdam (1967) \ref\no2
  • Th. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $\rz^N$ , Arch. Rational Mech. Anal., 124 (1993), 261–276 \ref\no3
  • J. Chabrowski, Variational Methods For Potential Operator Equations, W. de Gruyter, Studies in Mathematics, 24 , Berlin–New York (1997) \ref\no4
  • R. Chiappinelli, On spectral asymptotics and bifurcation for elliptic operators with odd superlinear term , Nonlinear Anal., 13 (1989), 871 – 878 \ref\no5
  • C. V. Coffman, A minimum-maximum principle for a class of non-linear integral equations , J. Anal. Math., 22 (1969), 391–419 \ref\no6 ––––, On variational principles for sublinear boundary value problems , J. Differential Equations, 17 (1975), 26–60 \ref\no7 –––– \paper \ls theory, complementary principles and the Morse index, Nonlinear Anal., 12 (1988), 507–529 \ref\no8
  • M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh (1973) \ref\no9
  • M. Heid, Eine Methode zur Konstruktion von Lösungen semilinearer Gleichungen bei Vorgabe geometrischer Daten, Ph. D. Thesis, Johannes Gutenberg-Universität Mainz (1998) \ref\no10
  • M. Heid and H.-P. Heinz, Periodic solutions with prescribed properties for semilinear second-order equations and systems (Ladde, G. S. and Sambandham, M., Proc. of “Dynamic Systems and Applications", eds.), 2 , Dynamic Publishers Inc., Atlanta, Georgia (1996), 249–254 \ref\no11
  • H.-P. Heinz, Un principe de maximum-minimum pour les valeurs critiques d'une fonctionelle non linéaire , C. R. Acad. Sci. Paris Sér I, 275 (1972), 1317–1318 \ref\no12 ––––, Nodal properties and variational characterizations of solutions to nonlinear Sturm–Liouville problems , J. Differential Equations, 62 (1986), 299–333 \ref\no13 ––––, Free \ls theory and the bifurcation diagrams of certain singular nonlinear problems, J. Differential Equations, 66 (1987), 263–300 \ref\no14
  • D. Husemoller, Fibre Bundles, McGraw–Hill, New York (1966) \ref\no15
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York (1989) \ref\no16
  • E. Michael, Topologies on spaces of subsets , Trans. Amer. Math. Soc., 71 (1951), 152–182 \ref\no17
  • Z. Nehari, Characteristic values associated with a class of nonlinear second-order differential equations , Acta Math., 105 (1961), 141–175 \ref\no18
  • P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems , J. Funct. Anal., 7 (1971), 487–513 \ref\no19
  • P. H. Rabinowitz, Minimax Methods In Critical Point Theory With Applications To Differential Equations, Providence, Rh. Island (1986) \ref\no20
  • T. Shibata, Nodal and asymptotic properties of solutions to nonlinear elliptic eigenvalue problems on general level sets , Israel J. Math., 74 ; No. 2-3 (1991), 225–240 \ref\no21
  • M. Struwe, Infinitely many solutions of superlinear boundary value problems with rotational symmetry , Arch. Math., 36 (1981), 360–369 \ref\no22 ––––, Variational Methods-Applications To Nonlinear Partial Differential Equations And Hamiltonian Systems, 2nd edition, Springer-Verlag, Berlin-Heidelberg (1996) \ref\no23
  • M. Willem, Minimax Theorems, Birkhäuser, Boston-Basel-Berlin (1996) \ref\no24
  • E. Zeidler, Nonlinear Functional Analysis And Its Applications, Vol. III: Variational Methods And Optimization, Springer-Verlag, New York (1986) \ref\no25 ––––, Ljusternik–Schnirelman theory on general level sets , Math. Nachr., 129 (1986), 235–259