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 Mat. Sb. (N.S.), 1983, Volume 120(162), Number 3, Pages 354–370 (Mi msb2135)

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

V. R. Kardashov

Abstract: 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.

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English version:
Mathematics of the USSR-Sbornik, 1984, 48:2, 349–363

Bibliographic databases:

UDC: 517.944
MSC: Primary 47H12, 47H15; Secondary 46E35, 55M30, 58B15, 58C40, 58E05

Citation: V. R. Kardashov, “Structure of the spectrum and estimates for the eigenvalues of nonlinear homogeneous operators”, Mat. Sb. (N.S.), 120(162):3 (1983), 354–370; Math. USSR-Sb., 48:2 (1984), 349–363

Citation in format AMSBIB
\Bibitem{Kar83} \by V.~R.~Kardashov \paper Structure of the spectrum and estimates for the eigenvalues of nonlinear homogeneous operators \jour Mat. Sb. (N.S.) \yr 1983 \vol 120(162) \issue 3 \pages 354--370 \mathnet{http://mi.mathnet.ru/msb2135} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=691983} \zmath{https://zbmath.org/?q=an:0558.47044|0525.47044} \transl \jour Math. USSR-Sb. \yr 1984 \vol 48 \issue 2 \pages 349--363 \crossref{https://doi.org/10.1070/SM1984v048n02ABEH002679}