arXiv subjects
Publications and source records attributed to Fabiana Leoni.
For the fully nonlinear stationary logistic equation ${\mathcal F}(x,D^2u)+μu=k(x)u^p$ with $p>1$ and $k(x)\geq 0$, in a bounded domain with Dirichlet boundary condition, we determine, in terms of $μ$, the existence and uniqueness or the nonexistence of a positive solution. Furthermore, we study the asymptotic behavior of the solutions when $μ$ approaches the boundary points of the existence range.
We consider equations involving the truncated laplacians and having lower order terms with singular potentials posed in punctured balls. We study both the principal eigenvalue problem and the problem of classification of solutions, in dependence of their asymptotic behaviour near the origin, for equations having also superlinear absorbing lower order terms. In the case of the maximising truncated Laplacian "Pk+", owing to the mild degeneracy of the operator, we obtain results which are analogous to the results for the Laplacian in dimension k. On the other hand, for minimising operator "Pk-" we show that the strong degeneracy in ellipticity of the operator produces radically different results.
We study fully nonlinear uniformly elliptic equations having a singular reaction term with inverse quadratic potential and an absorbing superlinear term of p-power type. We consider equations posed in punctured balls centered at the origin, and we prove that all radial solutions are singular around the origin, by providing a complete classification in dependence of p of their asymptotic behavior near the singularity.
This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions $( \barλ_γ, u_γ)$ of the equation $$F( D^2 u_γ)+ \bar λ_γ\frac{u_γ}{r^γ} = 0\ {\rm in} \ B(0,1)\setminus \{0\}, \ u_γ= 0 \ {\rm on} \ \partial B(0,1)$$ where $u_γ>0$ in $B(0,1)\setminus \{0\}$, and $γ>0$. We prove existence of radial solutions which are continuous on $\overline{ B(0,1)}$ in the case $γ<2$, existence of unbounded solutions in the case $γ= 2$ and a non existence result for $γ>2$. We also give the explicit value of $\bar λ_2$ in the case of Pucci's operators, which generalizes the Hardy--Sobolev constant for the Laplacian.
We prove existence, uniqueness and regularity results for mixed boundary value problems associated with fully nonlinear, possibly singular or degenerate elliptic equations. Our main result is a global Hölder estimate for solutions, obtained by means of the comparison principle and the construction of ad hoc barriers. The global Hölder estimate immediately yields a compactness result in the space of solutions, which could be applied in the study of principal eigenvalues and principal eigenfunctions of mixed boundary value problems.
We give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in $\mathbb R^N$, and equipped with homogeneous Dirichlet boundary conditions.
We study radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values, showing that new concentration phenomena occur. Finally we define a suitable weighted energy for these solutions and compute its limit value.
We prove a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators. We then apply the result in order to obtain gradient boundary blow up rates for ergodic functions in bounded domains related to degenerate/singular operators, and, as a further consequence, we deduce the uniqueness of the ergodic functions.
In this note, we prove $\mathcal{C}^{1,γ}$ regularity for solutions of some fully nonlinear degenerate elliptic equations with "superlinear" and "subquadratic " Hamiltonian terms. As an application, we complete the results of \cite{BDL1} concerning the associated ergodic problem, proving, among other facts, the uniqueness, up to constants, of the ergodic function.
For a class of fully nonlinear equations having second order operators which may be singular or degenerate when the gradient of the solutions vanishes, and having first order terms with power growth, we prove the existence and uniqueness of suitably defined viscosity solution of Dirichlet problem and we further show that it is a Lipschitz continuous function.
We study the asymptotic behaviour of positive solutions of fully nonlinear elliptic equations in a ball, as the exponent of the power nonlinearity approaches a critical value. We show that solutions concentrate and blow up at the center of the ball, while a suitable associated energy remains invariant.
We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when the gradient of solutions vanishes. We prove the convergence of both explosive solutions and solutions of Dirichlet problems for approximating equations. We further characterize the ergodic constant as the infimum of constants for which there exist bounded sub solutions. As intermediate results of independent interest, we prove a priori Lipschitz estimates depending only on the norm of the zeroth order term, and a comparison principle for equations having no zero order terms.
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.
We derive explicit expressions of the homogeneous solutions in two dimensional cones for Pucci's extremal equations. As examples of possible applications, we obtain monotonicity formulas for all nonnegative supersolutions and necessary and sufficient explicit conditions for non--existence results of Liouville type.
We consider the fully nonlinear problem \begin{equation*} \begin{cases} -F(x,D^2u)=|u|^{p-1}u & \text{in $Ω$}\\ u=0 & \text{on $\partialΩ$} \end{cases} \end{equation*} where $F$ is uniformly elliptic, $p>1$ and $Ω$ 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 $Ω$ 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 $Ω$ 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, $Ω$ is a ball and $p$ is subcritical.
We study symmetry properties of viscosity solutions of fully nonlinear uniformly elliptic equations. We show that if $u$ is a viscosity solution of a rotationally invariant equation of the form $F(x,D^2u)+f(x,u)=0$, then the operator $\mathcal{L}_u=\mathcal{M}^++\frac{\partial f}{\partial u}(x,u)$, where $\mathcal{M}^+$ is the Pucci's sup--operator, plays the role of the linearized operator at $u$. In particular, we prove that if $u$ is a solution in a radial bounded domain, if $f$ is convex in $u$ and if the principal eigenvalue of $\mathcal{L}_u$ (associated with positive eigenfunctions) in any half domain is nonnegative, then $u$ is foliated Schwarz symmetric. We apply our symmetry results to obtain bounds on the spectrum and to deduce properties of possible nodal eigenfunctions for the operator $\mathcal{M}^+$.
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.
We consider fully nonlinear degenerate elliptic equations with zero and first order terms. We provide a priori upper bounds and characterize the existence of entire subsolutions under growth conditions on the lower order coefficients which extend the classical Keller--Osserman condition for semilinear equations.