01-142 Michael Heid, Hans-Peter Heinz, Tobias Weth

Nonlinear Eigenvalue Problems Of Schrödinger Type Admitting Eigenfunctions With Given Spectral Characteristics (78K, LaTeX 2e) Apr 11, 01

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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.

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