Nonlinear eigenvalue problems admitting eigenfunctions with known geometric properties

Michael Heid, Hans-Peter Heinz

DOI: http://dx.doi.org/10.12775/TMNA.1999.002

Abstract


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.

Keywords


Ljusternik-Schnirelman levels; Krasnosel'skiĭ genus; nodal properties o solutions; nonlinear Sturm-Liouville problems; nonlinear Hill's equation; semilinear second-order systems; periodic solutions; Morse index

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