### Nonlinear eigenvalue problems admitting eigenfunctions with known geometric properties

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.

$$

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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