Some existence results for a Paneitz type problem via the theory of critical points at infinity

In this paper a fourth order equation involving critical growth is considered under the Navier boundary condition: ${\mathrm{\Delta }}^{2}u=K{u}^p$, $u>0$ in Ω, $u=\mathrm{\Delta }u=0$ on ∂Ω, where K is a positive function, Ω is a bounded smooth domain in ${\mathbb{R}}^n$, $n⩾5$ and $p+1=2n/(n-4)$, is the critical Sobolev exponent. We give some topological conditions on K to ensure the existence of solution. Our methods involve the study of the critical points at infinity and their contribution to the topology of the level sets of the associated Euler–Lagrange functional.     

A Hardy type inequality for W02,1(Ω) functions

We consider functions $u\in W_0^{2,1}(\Omega )$, where $\Omega ?{\mathbb{R}}^N$ is a smooth bounded domain. We prove that $\frac{u(x)}{d(x)}\in W_0^{1,1}(\Omega )$ with$6\mathrm{?}\left(\frac{u(x)}{d(x)}\right)6_{L^1(\Omega )}?C6u{6}_{W^{2,1}(\Omega )},$ where d is a smooth positive function which coincides with $\mathrm{dist}(x,?\Omega )$ near ?Ω and C is a constant depending only on d and Ω.     

Optimal Wloc2,2-regularity,Pohozhaevʼs identity,and nonexistence of weak solutions to some quasilinear elliptic equations

We begin by establishing a sharp (optimal) $W_{\mathrm{loc}}^{2,2}$-regularity result for bounded weak solutions to a nonlinear elliptic equation with the p-Laplacian, ${\mathrm{\Delta }}_{p}u\stackrel{\mathrm{def}}{=}\mathrm{div}({|\mathrm{?}u|}^{p?2}\mathrm{?}u)$, $1<p<\infty$. We develop very precise, optimal regularity estimates on the ellipticity of this degenerate (for $2<p<\infty$) or singular (for $1<p<2$) problem. We apply this regularity result to prove Pohozhaev?s identity for a weak solution $u\in W^{1,p}(\Omega )$ of the elliptic Neumann problem(P)$?{\mathrm{\Delta }}_{p}u+W^{\prime }(u)=f(x)\text{in}\Omega ;?u/?\nu =0\text{on}?\Omega .$ Here, Ω is a bounded domain in ${\mathbb{R}}^N$ whose boundary ?Ω is a $C^2$-manifold, $\nu \equiv \nu (x_0)$ denotes the outer unit normal to ?Ω at $x_0\in ?\Omega$, $x=(x_1,\dots ,x_N)$ is a generic point in Ω, and $f\in L^{\infty }(\Omega )\cap W^{1,1}(\Omega )$. The potential $W:\mathbb{R}\to \mathbb{R}$ is assumed to be of class $C^1$ and of the typical double-well shape of type $W(s)={|1?{|s|}^{\beta }|}^{\alpha }$ for $s\in \mathbb{R}$, where $\alpha ,\beta >1$ are some constants. Finally, we take an advantage of the Pohozhaev identity to show that problem (P) with $f\equiv 0$ in Ω has no phase transition solution $u\in W^{1,p}(\Omega )$ ($1<p?N$), such that $?1?u?1$ in Ω with $u\equiv ?1$ in ${\Omega }_{?1}$ and $u\equiv 1$ in ${\Omega }_1$, where both ${\Omega }_{?1}$ and ${\Omega }_1$ are some nonempty subdomains of Ω. Such a scenario for u is possible only if $N=1$ and ${\Omega }_{?1}$, ${\Omega }_1$ are finite unions of suitable subintervals of the open interval $\Omega ?{\mathbb{R}}^1$.     

Large solutions to the Monge–Ampère equations with nonlinear gradient terms: Existence and boundary behavior

In this paper, we obtain conditions about the existence and boundary behavior of (strictly) convex solutions to the Monge–Ampère equations with boundary blow-up

$\det ?D^{2}u(x)=b(x)f(u(x))\pm |\mathrm{?}u{|}^q,x\in \mathrm{\Omega },u{|}_{?\mathrm{\Omega }}=+\infty ,$

and

$\det ?D^{2}u(x)=b(x)f(u(x))(1+|\mathrm{?}u{|}^q),x\in \mathrm{\Omega },u{|}_{?\mathrm{\Omega }}=+\infty ,$

where Ω is a strictly convex, bounded smooth domain in ${\mathbb{R}}^N$ with $N\ge 2$, $q\in [0,N]$ (or $q\in [0,N)$), $b\in C^{\infty }(\mathrm{\Omega })$ which is positive in Ω, but may vanish or blow up on the boundary, $f\in C[0,\infty )$, $f(0)=0$, and f is strictly increasing on $[0,\infty )$ (or $f\in C(\mathbb{R})$, $f(s)>0,?s\in \mathbb{R}$, and f is strictly increasing on $\mathbb{R}$).     

Analysis of some injection bounds for Sobolev spaces by wavelet decomposition

We consider the Sobolev spaces $H^s(\Omega )$ and $H_0^s(\Omega )$ and the Besov spaces $B_{2,\infty }^{1/2}(\Omega )$, where Ω is a sufficiently regular (see Lemma 2) subdomain of ${\mathbb{R}}^2$. It is well known that for the values of $s\in [0,1/2)$ the two Sobolev spaces coincide, with equivalence of the norms, and that the inclusion $B_{2,\infty }^{1/2}(\Omega )?H^s(\Omega )$ holds. The Note is concerned with the explicit analysis of the constants appearing in the continuity bounds for the injections $H^s(\Omega )?H_0^s(\Omega )$ and $B_{2,\infty }^{1/2}(\Omega )?H^s(\Omega )$ and of their dependence on the regularity s of the spaces. The analysis is carried out by using the wavelet characterization of the corresponding norms.     

Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms

In this Note we prove an a priori estimate and the existence of a solution for a class of nonlinear elliptic problems whose model is $?\mathrm{div}A(x)Du+{\alpha }_{0}u=\gamma {|Du|}^q+f(x)$, when $1<q<2$ and $f\in L^m(\Omega )$ for some suitable m. The main interest of the result lies in the a priori estimate, the complete proof of which is given in the Note. To cite this article: N. Grenon et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Radial symmetry of positive solutions for semilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry

Let $n\in \mathbb{N}$ with $n?2$, $a\in (?1,0)\cup (0,1]$ and $f:(0,1)\times (0,\infty )\to \mathbb{R}$ such that for each $u\in (0,\infty )$, $r?{(1+a{r}^2)}^{(n+2)/2}f(r,{(1+a{r}^2)}^{?(n?2)/2}u):(0,1)\to \mathbb{R}$ is nonincreasing. We show that each positive solution of$\mathrm{\Delta }u+f(|x|,u)=0\text{in}B,u=0\text{on}?B$ is radially symmetric, where B is the open unit ball in ${\mathbb{R}}^N$.     

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field $\mathbf{e}:=\frac{1}{2}(\mathbf{?}{\mathit{u}}^T+\mathbf{?}\mathit{u})\in {\mathbb{L}}^2(\Omega )$ can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain $\Omega ?{\mathbb{R}}^3$, instead of the displacement vector field $\mathit{u}\in {\mathit{H}}^1(\Omega )$ in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition $\mathit{u}=\mathbf{0}$ on a portion ${\Gamma }_0$ of the boundary of Ω can be recast, again as boundary conditions on ${\Gamma }_0$, but this time expressed only in terms of the new unknown $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.     

On a nonlinear Schrödinger equation with a localizing effect

We consider the nonlinear Schrödinger equation associated to a singular potential of the form $a{|u|}^{?(1?m)}u+bu$, for some $m\in (0,1)$, on a possible unbounded domain. We use some suitable energy methods to prove that if $\mathrm{Re}(a)+\mathrm{Im}(a)>0$ and if the initial and right hand side data have compact support then any possible solution must also have a compact support for any $t>0$. This property contrasts with the behavior of solutions associated to regular potentials $(m?1)$. Related results are proved also for the associated stationary problem and for self-similar solution on the whole space and potential $a{|u|}^{?(1?m)}u$. The existence of solutions is obtained by some compactness methods under additional conditions. To cite this article: P. Bégout, J.I. Díaz, C. R. Acad. Sci. Paris, Ser. I 342 (2006).