Michael Heid, Hans-Peter Heinz, Tobias Weth
Nonlinear Eigenvalue Problems Of Schrödinger Type Admitting
Eigenfunctions With Given Spectral Characteristics
(78K, LaTeX 2e)
ABSTRACT. The following work is an extension of our recent paper \cite{HdHz99}. We
still deal with nonlinear eigenvalue problems of the form
\begin{eqspeclab}{*}
A_0 y + B(y) y = \lambda y
\end{eqspeclab}
in a real Hilbert space $\cH$ with a semi-bounded self-adjoint operator
$A_0$, while for every y from a dense subspace $X$ of $\cH$, $B(y)$ is a
symmetric operator. The left--hand side is assumed to be related to a certain
auxiliary functional $\psi$, and the associated linear problems
\begin{eqspeclab}{**}
A_0 v + B(y) v = \mu v
\end{eqspeclab}
are supposed to have non-empty discrete spectrum $\: (y \in X)$.We
reformulate and generalize the topological method presented by the authors in
$\cite{HdHz99}$ to construct solutions of (*) on a sphere $S_R := \{ y \in X
| \: \|y\|_{\cH} = R\}$ whose $\psi$-value is the $n$-th \ls level of $\psi
|_{S_R}$ and whose corresponding eigenvalue is the $n$-th eigenvalue of the
associated linear problem (**), where $R > 0$ and $n \in \nz$ 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 elliptic partial differential equations with radial
symmetry.