Principal eigenvalues for Fully Non Linear singular or degenerate operators in punctured balls

Abstract

This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear degenerate or singular uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions $({\bar{\lambda }}_{\gamma },u_{\gamma })$ of the equation ${|\nabla u|}^{\alpha }F\left(D^2{u}_{\gamma }\right)+{\bar{\lambda }}_{\gamma }\frac{u_{\gamma }^{1+\alpha }}{r^{\gamma }}=0\text{in}B(0,1)\setminus \left\{0\right\},u_{\gamma }=0\text{on}\partial B(0,1)$ where $u_{\gamma }>0$ in $B(0,1)$, $\alpha >-1$ and $\gamma >0$. We prove existence of radial solutions which are continuous on $\overline{B(0,1)}$ in the case $\gamma <2+\alpha$, and a non existence result for $\gamma >2+\alpha$. We also give the explicit value of ${\bar{\lambda }}_{2+\alpha }$ in the case of the Pucci’s operators, which generalizes the Hardy–Sobolev constant for the Laplacian, and the previous results of Birindelli et al. [1].