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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 $\lambda\in{\Bbb R}$ such that the maximum principle holds. The authors extend this view to a wide class of operators of the form $F(\nabla u,D^2 u)$. Let $\lambda_0$ be defined through the supremum of all $\lambda\in{\Bbb R}$ such that there exists $\Phi>0$ in $\Omega$ such that $F(\nabla\Phi,D^2\Phi)+\lambda\Phi^ {\alpha+1}\le 0$ 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 $F(\nabla u,D^2u)+\lambda|u|^\alpha u$ satisfies the maximum principle in $\Omega$ if $\lambda<\lambda_0$.

35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
35B50Maximum principles (PDE)
35B65Smoothness and regularity of solutions of PDE
35P15Estimation of eigenvalues and upper and lower bounds for PD operators