Principal eigenvalue of a very badly degenerate operator and applications. (English) Zbl 1132.35066
The author considers the eigenvalue problem
\[
-\Delta_{\infty}u=\lambda u\;{\text{in}} \;\Omega,\qquad u=0\;{\text{on}} \;\partial\Omega\,,
\]
where \(\Omega\) is a bounded open subset of \(\mathbb R^{n}\), \(u\) is a function of \(\Omega\) to \(\mathbb R\), \(\lambda\) plays the role of an eigenvalue, and
\[ \Delta_{\infty}u\equiv \biggl(D^{2}u \frac{Du}{| Du| }\biggr) \frac{Du}{| Du| }. \]
It is proved that the least eigenvalue is positive and is delivered by the formula
\[ \lambda_{1}=\sup\{\lambda:\exists v>0\text{ in }\Omega,\text{ such that } -\Delta_{\infty}v\geq\lambda v\}. \]
The author also proves that \(\lambda_{1}\) admits a positive eigenfunction and that it can be characterized as the supremum of the values \(\lambda\) for which \(\Delta_{\infty}+\lambda I\) satisfies the maximum principle. Then some applications are presented to the analysis of the problem
\[ -\Delta_{\infty}\varphi= \lambda\varphi +f \quad \text{in }\Omega, \]
with respect to unknown \(\varphi\) for some positive function \(f\) of \(\Omega\) to \(\mathbb R\).
\[ \Delta_{\infty}u\equiv \biggl(D^{2}u \frac{Du}{| Du| }\biggr) \frac{Du}{| Du| }. \]
It is proved that the least eigenvalue is positive and is delivered by the formula
\[ \lambda_{1}=\sup\{\lambda:\exists v>0\text{ in }\Omega,\text{ such that } -\Delta_{\infty}v\geq\lambda v\}. \]
The author also proves that \(\lambda_{1}\) admits a positive eigenfunction and that it can be characterized as the supremum of the values \(\lambda\) for which \(\Delta_{\infty}+\lambda I\) satisfies the maximum principle. Then some applications are presented to the analysis of the problem
\[ -\Delta_{\infty}\varphi= \lambda\varphi +f \quad \text{in }\Omega, \]
with respect to unknown \(\varphi\) for some positive function \(f\) of \(\Omega\) to \(\mathbb R\).
Reviewer: Massimo Lanza de Cristoforis (Padova)
MSC:
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 35J60 | Nonlinear elliptic equations |
| 35J70 | Degenerate elliptic equations |
| 47J10 | Nonlinear spectral theory, nonlinear eigenvalue problems |
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