Symmetry minimizes the principal eigenvalue: an example for the Pucci's sup operator

We explicitly evaluate the principal eigenvalue of the extremal Pucci's sup--operator for a class of special plane domains, and we prove that, for fixed area, the eigenvalue is minimal for the most symmetric set.Comment: 11 pages, 7 figure

Liouville theorems for a family of very degenerate elliptic non linear operators

We prove nonexistence results of Liouville type for nonnegative viscosity solutions of some equations involving the fully nonlinear degenerate elliptic operators ${\cal P}^\pm_k$, defined respectively as the sum of the largest and the smallest $k$ eigenvalues of the Hessian matrix. For the operator ${\cal P}^+_k$ we obtain results analogous to those which hold for the Laplace operator in space dimension $k$. Whereas, owing to the stronger degeneracy of the operator ${\cal P}^-_k$, we get totally different results.Comment: 15 page

Existence results for fully nonlinear equations in radial domains

We consider the fully nonlinear problem \begin{equation*} \begin{cases} -F(x,D^2u)=|u|^{p-1}u & \text{in $\Omega$}\\ u=0 & \text{on $\partial\Omega$} \end{cases} \end{equation*} where $F$ is uniformly elliptic, $p>1$ and $\Omega$ is either an annulus or a ball in \Rn, $n\geq2$. \\ We prove the following results: \begin{itemize} \item[i)] existence of a positive/negative radial solution for every exponent $p>1$, if $\Omega$ is an annulus; \item[ii)] existence of infinitely many sign changing radial solutions for every $p>1$, characterized by the number of nodal regions, if $\Omega$ is an annulus; \item[iii)] existence of infinitely many sign changing radial solutions characterized by the number of nodal regions, if $F$ is one of the Pucci's operator, $\Omega$ is a ball and $p$ is subcritical.Comment: 19 pages, 0 figure

Nuovi fenomeni di concentrazione per soluzioni radiali di segno variabile di equazioni ellittiche completamente non lineari

We present recent results about radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems posed in a ball, driven by the extremal Pucci's operators and provided with power zero order terms. We show that new critical exponents appear, related to the existence or nonexistence of sign-changing solutions and due to the fully nonlinear character of the considered problem. Furthermore, we analyze the new concentration phenomena occurring as the exponents approach the critical values.Vengono presentati alcuni risultati recenti riguardanti soluzioni radiali di segno variabile per una classe di problemi di Dirichlet completamente non lineari, posti in domini sferici, aventi gli operatori estremali di Pucci come parte principale e termini di ordine zero di tipo potenza. Mostreremo come l'esistenza o non esistenza di soluzioni sia regolata da nuovi esponenti critici tipici del carattere completamente non lineare del problema considerato. Analizzeremo inoltre i nuovi fenomeni di concentrazione che si verificano quando gli esponenti convergono ai valori critici.&nbsp

Explicit subsolutions and a Liouville theorem for fully nonlinear uniformly elliptic inequalities in halfspaces

We prove a Liouville type theorem for arbitrarily growing positive viscosity supersolutions of fully nonlinear uniformly elliptic equations in halfspaces. Precisely, let $M^-_{\lambda, \Lambda}$ be the Pucci's inf- operator, defined as the infimum of all linear uniformly elliptic operators with ellipticity constants $\Lambda \geq \lambda >0$. Then, we prove that the inequality $M^-_{\lambda, \Lambda}(D^2u) +u^p \leq 0$ does not have any positive viscosity solution in a halfspace provided that $-1\leq p \leq (\Lambda/\lambda n+1)/(\Lambda/\lambda n-1)$, whereas positive solutions do exist if either $p < -1$ or $p > (\Lambda/\lambda (n-1)+2)/(\Lambda/\lambda (n-1))$. This will be accomplished by constructing explicit subsolutions of the homogeneous equation $M^-_{\lambda, \Lambda}(D^2u)=0$ and by proving a nonlinear version in a halfspace of the classical Hadamard three-circles Theorem for entire superharmonic functions.Comment: 18 page

Radial solutions of Lane-Emden-Fowler equations with Pucci's extremal operators

We report on some recent results obtained for positive radial solutions of Lane-Emden-Fowler type equations with Pucci's operators as principal parts. The presented results include the asymptotic analysis of almost critical solutions in the unit ball,  existence results in annular domains and sharp Liouville-type results for exterior Dirichlet problems.

Entire subsolutions of fully nonlinear degenerate elliptic equations

We prove existence and non existence results for fully nonlinear degenerate elliptic inequalities, by showing that the classical Keller--Osserman condition on the zero order term is a necessary and sufficient condition for the existence of entire sub solutions