Generalized eigenvalues for fully nonlinear singular or degenerate operators in the radial case. (English) Zbl 1191.35200

Demengel, Françoise

Adv. Differ. Equ. 14, No. 11-12, 1127-1154 (2009).

In this article, the generalized eigenvalues for general operators fully nonlinear singular or degenerate homogeneous of degree \(1+\alpha\) with \(\alpha>-1\) are considered on a bounded domain \(\Omega\) in \(\mathbb R^N\). Specifically, the radial case for the operator \(F(Du,D^2u)= |\nabla u|^\alpha {\mathcal M}_{a,A}(D^2u)\) is treated, where \({\mathcal M}_{a,A}\) is the Pucci’s operator \({\mathcal M}_{a,A} (M) = A \operatorname{tr}(M^+)- a \operatorname{tr}(M^-)\) with \(a \leq A\). The authors prove the existence of a denumerable set of eigenvalues which are simple and isolated, and some continuity results for the eigenvalues with respect to the parameters \(\alpha, a, A\).

Reviewer: Chie-Ping Chu (Taipei)

MSC:

35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J25	Boundary value problems for second-order elliptic equations
35J60	Nonlinear elliptic equations
35P15	Estimates of eigenvalues in context of PDEs
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

generalized eigenvalues; fully nonlinear singular or degenerate operators; radial case