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First eigenvalue and maximum principle for fully nonlinear singular operators. (English) Zbl 1132.35427
The authors give a definition of the first eigenvalue for a class of fully nonlinear elliptic operators which are nonvariational but homogeneous. The main idea is to exploit the property that an elliptic operator satisfies a maximum principle if the involved parameter is less than the first eigenvalue. In the linear case it is well known that for the operator $Mu:=-{\Delta }u+\lambda u$ the maximum principle holds if $\lambda <{\lambda }_{0}$, where ${\lambda }_{0}$ is the first eigenvalue of $-{\Delta }u=\lambda u$ in ${\Omega }$, $u=0$ on $\partial {\Omega }$. Thus, ${\lambda }_{0}$ is the supremum of all $\lambda \in ℝ$ such that the maximum principle holds. The authors extend this view to a wide class of operators of the form $F\left(\nabla u,{D}^{2}u\right)$. Let ${\lambda }_{0}$ be defined through the supremum of all $\lambda \in ℝ$ such that there exists ${\Phi }>0$ in ${\Omega }$ such that $F\left(\nabla {\Phi },{D}^{2}{\Phi }\right)+\lambda {{\Phi }}^{\alpha +1}\le 0$ in the viscosity sense, where $\alpha >-1$. The authors show under additional assumptions that this ${\lambda }_{0}$ is the first eigenvalue of $-F$ in ${\Omega }$ in the sense that $F\left(\nabla u,{D}^{2}u\right)+\lambda {|u|}^{\alpha }u$ satisfies the maximum principle in ${\Omega }$ if $\lambda <{\lambda }_{0}$.

##### MSC:
 35P30 Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory 35B50 Maximum principles (PDE) 35B65 Smoothness and regularity of solutions of PDE 35P15 Estimation of eigenvalues and upper and lower bounds for PD operators