Structure of the spectrum and estimates for the eigenvalues of nonlinear homogeneous operators

V. R. Kardashov

Sbornik. Mathematics, Tome 48 (1984) no. 2, pp. 349-363

Résumé

In this paper conditions are given for the spectrum in an eigenvalue problem of the form $$\lambda A(u)=B(u)$$ to be discrete, where $A$ and $B$ are operators that are odd-homogeneous of degree $p-1$ $(p\geqslant2)$, acting from a reflexive Banach space into the dual. It is proved that the eigenvalues vary monotonically as $A$ and $B$ vary in the normed linear space of homogeneous operators of degree $p-1$. Explicit formulas for the eigenvalues and functions are obtained for the case where $A$ and $B$ are the gradients of the norms in the spaces $W_p^1[\Omega_0]$ and $L_p[\Omega_0]$ ($\Omega_0$ is a parallelepiped in $E^m$). Using these formulas the author obtains estimates for the eigenvalues in homogeneous and asymptotically homogeneous problems with variable coefficients in the space $\overset{\circ}{W_p^1}[\Omega]$, where $\Omega$ is an arbitrary bounded domain in $E^m$. Bibliography: 12 titles.