arXiv subjects
Publications and source records attributed to Marie-Françoise Bidaut-Véron.
We study the local properties of positive solutions of the equation $-Δu+ m\abs{\nabla u}^q-e^{u}=0$ in a punctured domain $Ω\setminus\{0\}$ of $R^N$, $N\geq 2$, where $m$ is a positive parameter and $q>1$. We study particularly the existence of solutions with an isolated singularity and the local behaviour of such singular solutions. .
We study some local and global properties of solutions of $-Δu- m\abs{\nabla u}^q-e^{u}=0$ in a punctured domain $Ω\setminus\{0\}$, or in an exterior domain of $R^N$, $N\geq 2$, where $m$ is a positive parameter and $q>1$. We study particularly the local behaviour of solutions with an isolated singularity or the asymptotic behaviour for solutions defined in an exterior domain, and also the existence of solutions with the behaviours previously described. These behaviours change drastically according $q$ is smaller or larger than $2$. Many results are obtained by introducing various dynamical systems associated to the equation.
We study the local properties of positive solutions of the equation -Delta u= exp(u) in a punctured domain Omega of RN in the range of parameters q > 1 and M > 0. We prove a series of a priori estimates near a singular point. In the case of radial solutions we use various techniques inherited from the dynamical systems theory to obtain the precise behaviour of singular solutions. We prove also the existence of singular solutions with these precise behaviours.
We study the local properties of positive solutions of the equation $-Δu=u^p-m|\nabla u|^q$ in a punctured domain $Ω\setminus\{0\}$ of $\mathbb{R}^N$ or in a exterior domain $\mathbb{R}^N\setminus B_{r_0}$ in the range $\min\{p,q\}>1$ and $m>0$. We prove a series of a priori estimates depending $p$ and $q$, and of the sign of $q-\frac {2p}{p+1}$ and $q-p$. Using various techniques we obtain removability results for singular sets and we give a precise description of behaviour of solutions near an isolated singularity or at infinity in $\mathbb{R}^N$.
Here we study the solutions of any sign of the system --$Δ$u 1 = |$\nabla$u 2 | p , --$Δ$u 2 = |$\nabla$u 1 | q , in a domain of R N , N 3 and p, q > 0, pq > 1.. We show their relation with Lane-Emden Hardy-H{é}non equations --$Δ$ N p w= $ε$r $σ$ w q , $ε$ = $\pm$1, where u $\rightarrow$ $Δ$ N p u (p > 1) is the p-Laplacian in dimension N, q > p -- 1 and $σ$ $\in$ R. This leads us to explore these equations in not often tackled ranges of the parameters N, p, $σ$. We make a complete description of the radial solutions of the system and of the Hardy-Henon equations and give nonradial a priori estimates and Liouville type results.
We study properties of positive functions satisfying (E) --$Δ$u+m|$\nabla$u| q -- u p = 0 is a domain $Ω$ or in R N + when p > 1 and 1 < q < 2. We give sufficient conditions for the existence of a solution to (E) with a nonnegative measure $μ$ as boundary data, and these conditions are expressed in terms of Bessel capacities on the boundary. We also study removable boundary singularities and solutions with an isolated singularity on $\partial$$Ω$. The different results depends on two critical exponents for p = p c := N +1 N --1 and for q = q c := N +1 N and on the position of q with respect to 2p p+1. Contents
We study properties of nonnegative functions satisfying (E)$\;-Δu+u^p-M|\nabla u|^q=0$ is a domain of $\mathbb{R}^N$ when $p>1$, $M>0$ and $1<q<p$. We concentrate our analysis on the solutions of (E) with an isolated singularity, or in an exterior domain, or in the whole space. The existence of such solutions and their behaviours depend strongly on the values of the exponents $p$ and $q$ and in particular according to the sign of $q-\frac{2p}{p+1}$, and when $q=\frac{2p}{p+1}$, also on the value of the parameter $M$ which becomes a key element. The description of the different behaviours is made possible by a sharp analysis of the radial solutions of (E).
We study the equation --div(A(x, u)) = g(x, u, u) + $μ$ where $μ$ is a measure and either g(x, u, u) $\sim$ |u| q 1 u||u| q 2 or g(x, u, u) $\sim$ |u| s 1 u + ||u| s 2. We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity.
We study global properties of positive radial solutions of --$Δ$u = up +M |$\nabla$u|p+1 in RN wherep > 1 and M is a real number. We prove the existence or the non-existence of ground states and of solutions with singularity at 0 according to the values of M and p.
We study local and global properties of solutions of --$Δ$u = u p + M ||u| q in a domain $Ω$ of R N , in the range min{p, q} > 1 and M $\in$ R. We prove a priori estimates and existence or non-existence of ground states.
We study nonlinear systems of the form $-Δ\_pu=v^{q\_1}+μ,\;-Δ\_pv=u^{q\_2}+η$ and $F\_k[-u]=v^{s\_1}+μ,\;F\_k[-v]=u^{s\_2}+η$ in a bounded domain $Ω$ or in $\mathbb{R}^N$ where $μ$ and $η$ are nonnegative Radon measures, $Δ\_p$ and $F\_k$ are respectively the $p$-Laplacian and the $k$-Hessian operators and $q\_1$, $q\_2$, $s\_1$ and $s\_2$ positive numbers. We give necessary and sufficient conditions for existence expressed in terms of Riesz or Bessel capacities.
We study the existence of separable infinite harmonic functions in any cone of R N vanishing on its boundary under the form u(r, $σ$) = r --$β$ $ω$($σ$). We prove that such solutions exist, the spherical part $ω$ satisfies a nonlinear eigenvalue problem on a subdomain of the sphere S N --1 and that the exponents $β$ = $β$ + > 0 and $β$ = $β$ -- < 0 are uniquely determined if the domain is smooth.
We study the boundary behaviour of the solutions of (E) $-Δ_p u+|\nabla u|^q=0$ in a domain $Ω\subset \mathbb{R}^N$, when $N\geq p > q >p-1$. We show the existence of a critical exponent $q_* < p$ such that if $p-1 < q < q_*$ there exist positive solutions of (E) with an isolated singularity on $\partialΩ$ and that these solutions belong to two different classes of singular solutions. If $q_*\leq q < p$ no such solution exists and actually any boundary isolated singularity of a positive solution of (E) is removable. We prove that all the singular positive solutions are classified according the two types of singular solutions that we have constructed.
We give new criteria for the existence of weak solutions to an equation with a super linear source term \begin{align*}-Δu = u^q ~~\text{in}~Ω,~~u=σ~~\text{on }~\partialΩ\end{align*}where $Ω$ is a either a bounded smooth domain or $\mathbb{R}\_+^{N}$, $q\textgreater{}1$ and $σ\in \mathfrak{M}^+(\partialΩ)$ is a nonnegative Radon measure on $\partialΩ$. One of the criteria we obtain is expressed in terms of some Bessel capacities on $\partialΩ$. We also give a sufficient condition for the existence of weak solutions to equation with source mixed terms. \begin{align*} -Δu = |u|^{q\_1-1}u|\nabla u|^{q\_2} ~~\text{in}~Ω,~~u=σ~~\text{on }~\partialΩ\end{align*} where $q\_1,q\_2\geq 0, q\_1+q\_2\textgreater{}1, q\_2\textless{}2$, $σ\in \mathfrak{M}(\partialΩ)$ is a Radon measure on $\partialΩ$.
Here we study the initial trace problem for the nonnegative solutions of the equation \[ u\_{t}-Δu+|\nabla u|^{q}=0 \] in $Q\_{Ω,T}=Ω\times\left( 0,T\right) ,$ $T\leqq\infty,$ where $q>0,$ and $Ω=\mathbb{R}^{N},$ or $Ω$ is a smooth bounded domain of $\mathbb{R}^{N}$ and $u=0$ on $\partialΩ\times\left( 0,T\right) .$ We can define the trace at $t=0$ as a nonnegative Borel measure $(\mathcal{S} ,u\_{0}),$ where $S$ is the closed set where it is infinite, and $u\_{0}$ is a Radon measure on $Ω\backslash\mathcal{S}.$ We show that the trace is a Radon measure when $q\leqq1.$ For $q\in(1,(N+2)/(N+1)$ and any given Borel measure, we show the existence of a minimal solution, and a maximal one on conditions on $u\_{0}.$ When $\mathcal{S}$ $=\overlineω\capΩ$ and $ω$ is an open subset of $Ω,$ the existence extends to any $q\leqq2$ when $u\_{0}\in L\_{loc}^{1}(Ω)$ and any $q>1$ when $u\_{0}=0$. In particular there exists a self-similar nonradial solution with trace $(\mathbb{R}^{N+},0),$ with a growth rate of order $\left\vert x\right\vert ^{q^{\prime}}$ as $\left\vert x\right\vert \rightarrow\infty$ for fixed $t.$ Moreover we show that the solutions with trace $(\overlineω,0)$ in $Q\_{\mathbb{R}^{N},T}$ may present near $t=0$ a growth rate of order $t^{-1/(q-1)}$ in $ω$ and of order $t^{-(2-q)/(q-1)}$ on $\partial ω.$
Let $Ω$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=Ω\times(0,T).$ We study problems of the model type \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{Δ_{p}}u=μ\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partialΩ\times(0,T),\\ u(0)=u_{0}\qquad\text{in }Ω, \end{array} \right. \] where $p>1$, $μ\in\mathcal{M}_{b}(Q)$ and $u_{0}\in L^{1}(Ω).$ Our main result is a \textit{stability theorem }extending the results of Dal Maso, Murat, Orsina, Prignet, for the elliptic case, valid for quasilinear operators $u\longmapsto\mathcal{A}(u)=$div$(A(x,t,\nabla u))$\textit{. }
Let $Ω$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=Ω\times(0,T).$ We consider problems\textit{ }of the type % \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{Δ_{p}}u\pm\mathcal{G}(u)=μ\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partialΩ\times(0,T),\\ u(0)=u_{0}\qquad\text{in }Ω, \end{array} \right. \] where ${Δ_{p}}$ is the $p$-Laplacian, $μ$ is a bounded Radon measure, $u_{0}\in L^{1}(Ω),$ and $\pm\mathcal{G}(u)$ is an absorption or a source term$.$ In the model case $\mathcal{G}(u)=\pm\left\vert u\right\vert ^{q-1}u$ $(q>p-1),$ or $\mathcal{G}$ has an exponential type. We prove the existence of renormalized solutions for any measure $μ$ in the subcritical case, and give sufficient conditions for existence in the general case, when $μ$ is good in time and satisfies suitable capacitary conditions.
We obtain new a priori estimates for the nonnegative solutions of the equation \[ u_{t}-Δu+|\nabla u|^{q}=0 \] in $Q_{Ω,T}=Ω\times\left( 0,T\right) ,$ $T\leqq\infty,$ where $q>0,$ and $Ω=\mathbb{R}^{N},$ or $Ω$ is a smooth bounded domain of $\mathbb{R}^{N}$ and $u=0$ on $\partialΩ\times\left( 0,T\right) .$ In case $Ω=\mathbb{R}^{N},$ we show that any solution $u\in C^{2,1}(Q_{\mathbb{R}^{N},T})$ of equation (1.1) in $Q_{\mathbb{R}^{N} ,T}$ (in particular any weak solution if $q\leqq2),$ without condition as $\left\vert x\right\vert \rightarrow\infty,$ satisfies the universal estimate \[ \left\vert \nabla u(.,t)\right\vert ^{q}\leqq\frac{1}{q-1}\frac{u(.,t)}% {t},\qquad\text{in }Q_{\mathbb{R}^{N},T}. \] Moreover we prove that the growth of $u$ is limited by $C(t+t^{-1/(q-1}% )(1+\left\vert x\right\vert ^{q^{\prime}}),$ where $C$ depends on $u.$ We also give existence properties of solutions in $Q_{Ω,T},$ for initial data locally integrable or even unbounded Radon measures. We give a nonuniqueness result in case $q>2.$ Finally we show that besides the local regularizing effect of the heat equation, $u$ satisfies a second effect of type $L_{loc}^{R}% -L_{loc}^{\infty},$ due to the gradient term.