arXiv subjects

Françoise Demengel

Publications and source records attributed to Françoise Demengel.

7 recordsLinked to original sources

arXiv2025

Radial solutions of truncated Laplacian equations in punctured balls

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.

math.AP↗

arXiv2024

Radial singular solutions of fully nonlinear equations in punctured balls

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.

math.AP↗

arXiv2023

Principal eigenvalues for Fully non linear singular or degenerate operators in punctured balls

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 $( \barλ_γ, u_γ)$ of the equation $$| \nabla u |^αF( D^2 u_γ)+ \bar λ_γ{u_γ^{1+α} \over r^γ} = 0\ {\rm in} \ B(0,1)\setminus \{0\}, \ u_γ= 0 \ {\rm on} \ \partial B(0,1)$$ where $u_γ>0$ in $B(0,1)$, $α>-1$ and $γ>0$. We prove existence of radial solutions which are continuous on $\overline{ B(0,1)}$ in the case $γ<2+α$, existence of unbounded solutions which do ot satisfy the boundary condition in the case $γ= 2+α$ and a non existence result for $γ>2+α$. We also give the explicit value of $\bar λ_{2+α} $ in the case of Pucci's operators, which generalizes the Hardy--Sobolev constant for the Laplacian, and the previous results of Birindelli, Demengel and Leoni

math.AP↗

arXiv2023

Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls

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.

math.AP↗

arXiv2020

Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators

We study the ergodic problem for fully nonlinear elliptic operators $F( \nabla u, D^2 u)$ which may be degenerate when at least one of the components of the gradient vanishes. We extend here the results in the celebrated paper of Lasry and Lions, the ones of Leonori and Porretta Capuzzo Dolcetta Leoni and Porretta, Birindelli et al.

math.AP↗

arXiv2018

Extremal functions for an embedding from some anisotropic space, and partial differential equation involving the "one Laplacian"

In this paper, we prove the existence of extremal functions for the best constant of embedding from anisotropic space, allowing some of the Sobolev exponents to be equal to $1$. We prove also that the extremal functions satisfy a partial differential equation involving the $1$ Laplacian.

math.AP↗

arXiv2013

Qualitative properties of a nonlinear system involving the $p$-Laplacian operator

In this article we consider the nonlinear system involving the $p$-Laplacian $$\left\{\begin{array}{lc} |u^\prime |^{p-2} u^{\prime \prime} = u^{p-1} v^p& |v^\prime |^{p-2} v^{\prime \prime} = v^{p-1} u^p&\ {\rm on} \ \R, u\geq 0, v\geq 0& \end{array}\right.$$ for which we prove symmetry, asymptotic behavior and non degeneracy properties. This can help to a better understanding to what happens in the $N$ dimensional case, for which several authors prove a De Giorgi Type result under some additional growth and monotonicity assumptions.

math.AP↗