# Louis DupaigneClaude Bernard University Lyon 1 | UCBL · Institut Camille Jordan (ICJ)

Louis Dupaigne

PhD & HdR

## About

56

Publications

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2,339

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Citations since 2017

## Publications

Publications (56)

We prove that 0 the only classical solution of the Lane-Emden equation in the half-space which is stable outside a compact set. We also consider weak solutions and the case of general cones.

In this paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are stable or more generally of finite Morse index or even more generally locally stable.

We are interested in the Caffarelli-Kohn-Nirenberg inequality (CKN in short), introduced by these authors in 1984. We explain why the CKN inequality can be viewed as a Sobolev inequality on a weighted Riemannian manifold. More precisely, we prove that the CKN inequality can be interpreted in this way on three different and equivalent models, obtain...

We prove that the Dirichlet problem for the Lane–Emden equation in a half-space has no positive solution that is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution that is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a...

In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.

In this note we present a new proof of Sobolev's inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous inequality and to explain how our method, relying on a gradient-flow interpretation, is simple and robust. In p...

We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solution which is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution which is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such...

We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.

We study the Lane-Emden equation in strips.

We present a construction of harmonic functions on bounded domains for the
spectral fractional Laplacian operator and we classify them in terms of their
divergent profile at the boundary. This is used to establish and solve boundary
value problems associated with nonhomogeneous boundary conditions. We provide a
weak-$L^1$ theory to show how problem...

We classify solutions of finite Morse index of the fractional Lane- Emden
equation

We consider Liouville-type and partial regularity results for the
nonlinear fourth-order problem $$ \Delta^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$
where $ p>1$ and $n\ge1$. We give a complete classification of stable
and finite Morse index solutions (whether positive or sign changing), in
the full exponent range. We also compute an upper bound of the Hausd...

We study stable positive radially symmetric solutions for the Lane-Emden
system $-\Delta u=v^p$ in $\R^N$, $-\Delta v=u^q$ in $\R^N$, where $p,q\geq 1$.
We obtain a new critical curve that optimally describes the existence of such
solutions.

We study stable and finite Morse index solutions of the equation Δ2 = eu. If the equation is posed in ℝN, we classify radial stable solutions. We then construct nonradial stable solutions and we prove that, unlike the corresponding second order problem, no Liouville-type theorem holds, unless additional information is available on the asymptotics o...

In this short note, we study the smoothness of the extremal solutions to the
Liouville system

Given a nondecreasing nonlinearity $f$, we prove uniqueness of large
solutions in the following two cases: the domain is the ball or the domain has
nonnegative mean curvature and the nonlinearity is asymptotically convex.

We analyze the semilinear elliptic equation $\Delta u=\rho(x) f(u)$, $u>0$ in
${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative
study of entire large solutions, that is, solutions $u$ such that
$\lim_{|x|\rightarrow +\infty}u(x)=+\infty$. Assuming that $f$ satisfies the
Keller-Osserman growth assumption and that $\rho$ de...

We investigate stable solutions of elliptic equations of the type where n ≥ 2, s ∈ (0, 1), λ ≥0 and f is any smooth positive superlinear function. The operator (− Δ) stands for the fractional Laplacian, a pseudo-differential operator of order 2s. According to the value of λ, we study the existence and regularity of weak solutions u.

We prove regularity and partial regularity results for finite Morse index solutions u∈H1(Ω)∩Lp(Ω) to the Lane–Emden equation −Δu=|u|p−1u in Ω.

Stable solutions are ubiquitous in differential equations. They represent meaningful solutions from a physical point of view and appear in many applications, including mathematical physics (combustion, phase transition theory) and geometry (minimal surfaces). Stable Solutions of Elliptic Partial Differential Equations offers a self-contained presen...

In this article, we investigate a water wave model with a nonlocal viscous term ut + ux + βuxxx + √v/√π ∫ot ut(s)/√t-s ds + uux = vuxx. The wellposedness of the equation and the decay rate of solutions are investigated theoretically and numerically.

We investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right . \end{equation*} where $n\ge2$, $s \in (0,1)$, $\lambda \geq 0$ and $f$ is any smooth positive superlinear f...

We calculate the full asymptotic expansion of boundary blow-up solutions, for any nonlinearity f. Our approach enables us to state sharp qualitative results regarding uniqueness and ra-dial symmetry of solutions, as well as a characterization of nonlinearities for which the blow-up rate is universal. Lastly, we study in more detail the standard non...

Several Liouville-type theorems are presented for stable solutions of the equation -Δu=f(u) in ℝ N , where f>0 is a general convex, nondecreasing function. Extensions to solutions which are merely stable outside a compact set are discussed.

We prove a Liouville-type theorem for bounded stable solutions $v \in C^2(\R^n)$ of elliptic equations of the type (-\Delta)^s v= f(v)\qquad {in $\R^n$,} where $s \in (0,1)$ {and $f$ is any nonnegative function}. The operator $(-\Delta)^s$ stands for the fractional Laplacian, a pseudo-differential operator of symbol $|\xi |^{2s}$.

We study the existence, uniqueness and boundary profile of nonnegative boundary blow-up solution to the cooperative system in a smooth bounded domain of RN, where f, g are nondecreasing, nonnegative C1 functions vanishing in (−∞,0] and β>0 is a parameter.

We extend our recent results on the classification of stable solutions of the equation −Δu=f(u) in RN, where f≥0 is a general convex, non-decreasing function.

We calculate the full asymptotic expansion of boundary blow-up so- lutions, for any nonlinearity f . Our approach enables us to state sharp qualitative results regarding uniqueness and ra- dial symmetry of solutions, as well as a characterization of nonlinearities for which the blow-up rate is universal. At last, we study in more detail the standar...

We consider Delta u = 0 in Omega, partial derivative u/partial derivative v = lambda f(u) on Gamma(1), u= 0 on Gamma(2) where lambda > 0, f(u) = e(u) or f(u) = (1 + u)(p), Gamma(1), Gamma(2) is a partition of partial derivative Omega and Omega subset of R-N. We determine sharp conditions on the dimension N and p > 1 such that the extremal solution...

The paper proves Liouville-type results for stable solutions of semilinear elliptic PDEs with convex nonlinearity, posed on the entire Euclidean space. Extensions to solutions which are stable outside a compact set are also presented.

Let $\lambda^*>0$ denote the largest possible value of $\lambda$ such that \begin{align*} \left\{\begin{aligned} \Delta^2 u & = \la e^u && \text{in $B $} u &= \pd{u}{n} = 0 && \text{on $ \pa B $} \end{aligned} \right. \end{align*} has a solution, where $B$ is the unit ball in $\R^N$ and $n$ is the exterior unit normal vector. We show that for $\lam...

he equation $-\Delta u = \lambda e^u$ posed in the unit ball $B \subseteq \R^N$, with homogeneous Dirichlet condition $u|_{\partial B} = 0$, has the singular solution $U=\log\frac1{|x|^2}$ when $\lambda = 2(N-2)$. If $N\ge 4$ we show that under small deformations of the ball there is a singular solution $(u,\lambda)$ close to $(U,2(N-2))$. In dimen...

The equation -Δu = λe u posed in the unit ball B ⊆ ℝ N , with homogeneous Dirichlet condition u| ∂B = 0, has the singular solution [Formula: see text] when λ = 2(N - 2). If N ≥ 4 we show that under small deformations of the ball there is a singular solution (u,λ) close to (U,2(N - 2)). In dimension N ≥ 11 it corresponds to the extremal solution — t...

We study a one-dimensional non-local variant of Fisher's equation describing the spatial spread of a mutant in a given population, and its generalization to the so-called monostable nonlinearity. The dispersion of the genetic characters is assumed to follow a non-local diffusion law modelled by a convolution operator. We prove that, as in the class...

We are concerned with singular elliptic problems of the form −Δu±p(d(x))g(u)=λf(x,u)+μa|∇u| in Ω, where Ω is a smooth bounded domain in RN, d(x)=dist(x,∂Ω), λ>0, μ∈R, 0<a⩽2, and f is a nondecreasing function. We assume that p(d(x)) is a positive weight with possible singular behavior on the boundary of Ω and that the nonlinearity g is unbounded aro...

This article is concerned with the existence, uniqueness and numerical approximation of boundary blow up solutions for elliptic PDE's as $\Delta u=f(u)$ where $f$ satisfies the so-called Keller-Osserman condition. We {\bleu characterize} existence of such solutions {\bleu for non-monotone $f$} . As an example, we construct an infinite family of bou...

We study the existence of solutions of the nonlinear problem -Δu+g(u)=μinΩ,u=0on∂Ω,(0·1) where μ is a bounded measure and g is a continuous nondecreasing function such that g(0)=0. In this paper, we assume that the nonlinearity g satisfies lim t↑1 g(t)=+∞(0·2) Problem (0.1) need not have a solution for every measure μ. We prove that, given μ, there...

Borel summation techniques are developed to obtain exact invariants from formal adiabatic invariants (given as divergent series in a small parameter) for a class of differential equations, under assumptions of analyticity of the coefficients. The method relies on the study of associated partial differential equations in the complex plane. The type...

We are concerned with singular elliptic problems of the form $-\Delta u\pm p(d(x))g(u)=\la f(x,u)+\mu |\nabla u|^a$ in $\Omega,$ where $\Omega$ is a smooth bounded domain in $\RR^N$, $d(x)={\rm dist}(x,\partial\Omega),$ $\la>0,$ $\mu\in\RR$, $0<a\leq 2$, and $f,k$ are nonnegative and nondecreasing functions. We assume that $p(d(x))$ is a positive w...

On a semilinear elliptic equation with inverse square potential

Let $\\Omega\\subset\\Bbb{R}^N$ be a bounded domain and denote by ${\\rm cap}_2$ the standard $H^1$-capacity. For any Radon measure $µ$ in $\\Bbb{R}^N$, consider the \"Radon-Nikodym\" decomposition $µ=\\mu_{\\rm d}+\\mu_{\\rm c}$ with respect to ${\\rm cap}_2$, so that the diffuse measure $\\mu_{\\rm d}$ satisfies $\\mu_{\\rm d}(A)=0$ for any Borel...

In this well-written paper, the authors study operators of the form $L=-\\Delta -µd^{-2}$, where $d(x)={\\rm dist}(x,\\Sigma)$, $µ\\in R$ and $\\Sigma \\subset R^{n}$. More precisely, they study inequalities which suggest that the operator $L$ has a positive first eigenvalue. Such inequalities, with many variants of the parameters, have already bee...

The object of this paper is to provide variational formulas characterizing the speed of travelling front solutions of the following nonlocal diffusion equation:Where J is a dispersion kernel and f is any of the nonlinearities commonly used in various models ranging from combustion theory of ecology. In several situations, such as population dynamic...

We provide results of the existence, uniqueness and asymptotic behavior of travelling-wave solutions for convolution equations involving different kinds of nonlinearities (bistable, ignition and monostable). We recover for these equations most of the known results about the standard equation ∂u∂t+u″+f(u)=0. Some min–max formulas are also given. To...

Several comparison results are obtained for solutions to linear elliptic and parabolic equations with a singular potential. Solutions to these equations are singular in many cases, and our results roughly say that they all have comparable singularities, provided that they belong to an appropriate space. We formulate the hypothesis on the potential...

Comparison principles for PDEs with a singular potential

Semilinear elliptic PDEs with a singular potential

Jury: Président: H. Brezis, Membres : T. Cazenave, J.I. Diaz, J. Goldstein, F. Merle, F. Pacard, P. Souplet, F. Weissler, Q. Zhang

We generalize a Liouville-type theorem of Keller and Osserman for the semilinear Poisson equation -Δu+f(u)=0 in ℝ N and revisit a celebrated symmetry result of Gidas, Ni and Nirenberg.

"Graduate Program in Mathematics." Thesis (Ph. D.)--Rutgers University, 2002. Includes abstract. Vita. Includes bibliographical references.