Fucik spectra for vector equations

C. Fabry, Catholic University of Louvain, Belgium

E. J. Qualitative Theory of Diff. Equ., No. 7. (2000), pp. 1-24.

Abstract: Let $L:\hbox{dom} L\subset L^2(\Omega;R^N)\rightarrow L^2(\Omega;R^N)$ be a linear operator, $\Omega$ being open and bounded in $R^M$. The aim of this paper is to study the Fu\v c\'\i k spectrum for vector problems of the form $Lu=\alpha Au^+ -\beta Au^-$, where $A$ is an $N\times N$ matrix, $\alpha, \beta$ are real numbers, $u^+$ a vector defined componentwise by $(u^+)_i=\max\{u_i,0\}$, $u^-$ being defined similarly. With $\lambda^*$ an eigenvalue for the problem $Lu=\lambda Au$, we describe (locally) curves in the Fu\v c\'\i k spectrum passing through the point $(\lambda^*,\lambda^*)$, distinguishing different cases illustrated by examples, for which Fu\v c\'\i k curves have been computed numerically.

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