On the sub poly-harmonic property for solutions to (−Δ)pu < 0 in Rn

In this note, we mainly study the relation between the sign of ${(?\mathrm{\Delta })}^{p}u$ and ${(?\mathrm{\Delta })}^{p?i}u$ in ${\mathbb{R}}^n$ with $p?2$ and $n?2$ for $1?i?p?1$. Given the differential inequality ${(?\mathrm{\Delta })}^{p}u<0$, first we provide several sufficient conditions so that ${(?\mathrm{\Delta })}^{p?1}u<0$ holds. Then we provide conditions such that ${(?\mathrm{\Delta })}^{i}u<0$ for all $i=1,2,\dots ,p?1$, which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ${(?\mathrm{\Delta })}^{p}u={\mathrm{e}}^{2pu}$ and ${(?\mathrm{\Delta })}^{p}u=u^q$ with $q>0$ in ${\mathbb{R}}^n$.     

On the positive radial solutions of a class of singular semilinear elliptic equations

In this paper, we consider the following elliptic equation(0.1)$\mathit{div}\left(A\right(|x|\left)\mathrm{?}u\right)+B(|x|)u^p=0\text{in}{\mathbb{R}}^n,$ where $p>1$, $n?3$, $A(|x|)>0$ is differentiable in ${\mathbb{R}}^n?\{0\}$ and $B(|x|)$ is a given nonnegative Hölder continuous function in ${\mathbb{R}}^n?\{0\}$. The asymptotic behavior at infinity and structure of separation property of positive radial solutions with different initial data for (0.1) are discussed. Moreover, the existence and separation property of infinitely many positive solutions for Hardy equation and an equation related to Caffarelli–Kohn–Nirenberg inequality are obtained respectively, as special cases.     

Pointwise asymptotic behavior of modulated periodic reaction–diffusion waves

By working with the periodic resolvent kernel and the Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction–diffusion equations. With our linearized estimates together with a nonlinear iteration scheme developed by Johnson–Zumbrun, we obtain $L^p$-behavior ($p?1$) of a nonlinear solution to a perturbation equation of a reaction–diffusion equation with respect to initial data in $L^1\cap H^2$ recovering and slightly sharpening results obtained by Schneider using weighted energy and renormalization techniques. We obtain also pointwise nonlinear estimates with respect to two different initial perturbations $|u_0|?E_0{e}^{?{|x|}^2/M}$, ${|u_0|}_{H^2}?E_0$ and $|u_0|?E_0{(1+|x|)}^{?r}$, $r>2$, ${|u_0|}_{H^2}?E_0$ respectively, $E_0>0$ sufficiently small and $M>1$ sufficiently large, showing that behavior is that of a heat kernel. These pointwise bounds have not been obtained elsewhere, and do not appear to be accessible by previous techniques.     

Multiple positive solutions for semilinear elliptic systems

This article investigates the effect of the coefficient $f(z)$ of the critical nonlinearity. For sufficiently small $\lambda ,\mu >0$, there are at least k positive solutions of the semilinear elliptic systems$\{\begin{array}{c}?\mathrm{\Delta }u=\lambda g(z){|u|}^{p?2}u+\frac{\alpha }{\alpha +\beta }f(z){|u|}^{\alpha ?2}u{|v|}^{\beta }\text{in}\Omega ;\hfill \\ ?\mathrm{\Delta }v=\mu h(z){|v|}^{p?2}v+\frac{\beta }{\alpha +\beta }f(z){|u|}^{\alpha }{|v|}^{\beta ?2}v\text{in}\Omega ;\hfill \\ u=v=0\text{on}?\Omega ,\hfill \end{array}$ where $0\in \Omega ?{\mathbb{R}}^N$ is a bounded domain, $\alpha >1$, $\beta >1$ and $2<p<\alpha +\beta =2^{?}$ for $N>4$.     

On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states

Consider the Hénon equation with the homogeneous Neumann boundary condition

$?\mathrm{\Delta }u+u=|x{|}^{\alpha }{u}^p,u>0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{ on }?\mathrm{\Omega },$

where $\mathrm{\Omega }?B(0,1)?{\mathbb{R}}^N,N\ge 2$ and $?\mathrm{\Omega }\cap ?B(0,1)\ne ?$. We are concerned on the asymptotic behavior of ground state solutions as the parameter $\alpha \to \infty$. As $\alpha \to \infty$, the non-autonomous term $|x{|}^{\alpha }$ is getting singular near $|x|=1$. The singular behavior of $|x{|}^{\alpha }$ for large $\alpha >0$ forces the solution to blow up. Depending subtly on the $(N?1)?$dimensional measure $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$ and the nonlinear growth rate p, there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$. In particular, the critical exponent $2_{?}=\frac{2(N?1)}{N?2}$ for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any $p\in (1,2^{?}?1)$ and a smooth domain Ω.     

Harnack inequality for quasilinear elliptic equations on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equations

$\mathrm{div}\left(\frac{F^{\prime }(|\mathrm{?}u|)}{|\mathrm{?}u|}\mathrm{?}u\right)=h$

of p-Laplacian type on Riemannian manifolds, where an even function $F\in C^1\left(\mathbb{R}\right)\cap C^2(0,\infty )$ is supposed to be strictly convex on $(0,\infty )$. Under the assumption that either $F\in C^2(\mathbb{R})$ or its convex conjugate $F^{?}\in C^2(\mathbb{R})$ with some structural condition, we establish a (locally) uniform ABP type estimate and the Krylov–Safonov type Harnack inequality on Riemannian manifolds with the use of an intrinsic geometric quantity to the operator. Here, the $C^2$-regularities of F and $F^{?}$ account for degenerate and singular operators, respectively.     

Existence and concentration result for a class of fractional Kirchhoff equations with Hartree-type nonlinearities and steep potential well

In this paper, we study the following fractional Kirchhoff equations

$\{\begin{array}{c}(a+b{\int }_{{\mathbb{R}}^N}|{(?△)}^{\frac{\alpha }{2}}u{|}^2\mathrm{d}x){(?△)}^{\alpha }u+\lambda V(x)u=(|x{|}^{?\mu }?G(u))g(u),\hfill \\ u\in H^{\alpha }({\mathbb{R}}^N),N\ge 3,\hfill \end{array}$

where $a,b>0$ are constants, and ${(?△)}^{\alpha }$ is the fractional Laplacian operator with $\alpha \in (0,1),2<2_{\alpha ,\mu }^{?}=\frac{2N?\mu }{N?2\alpha }\le 2_{\alpha }^{?}=\frac{2N}{N?2\alpha }$, $0<\mu <2\alpha$, $\lambda >0$, is real parameter. $2_{\alpha }^{?}$ is the critical Sobolev exponent. g satisfies the Berestycki–Lions-type condition (see [2]). By using Poho?aev identity and concentration-compact theory, we show that the above problem has at least one nontrivial solution. Furthermore, the phenomenon of concentration of solutions is also explored. Our result supplements the results of Lü (see [8]) concerning the Hartree-type nonlinearity $g(u)=|u{|}^{p?1}u$ with $p\in (2,6?\alpha )$.     

Existence of standing waves of nonlinear Schrödinger equations with potentials vanishing at infinity

For a singularly perturbed nonlinear elliptic equation ${\epsilon }^2\mathrm{\Delta }u?V(x)u+u^p=0$, $x\in {\mathbb{R}}^N$, we prove the existence of bump solutions concentrating around positive critical points of V when nonnegative V is not identically zero for $p\in (\frac{N}{N?2},\frac{N+2}{N?2})$ or nonnegative V satisfies ${\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}_{|x|\to \infty }V(x){|x|}^2\log |x|>0$ for $p=\frac{N}{N?2}$.     

Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.     

Two remarks on liftings of maps with values into S1

Given a map $u\in L_{\mathrm{loc}}^1(\Omega ,S^1)$ with some regularity: ${|u|}_X=R<\infty$, we consider the problem of finding a lifting φ of u (i.e. a measurable function satisfying $u={\mathrm{e}}^{\mathrm{i}\phi }$) with the same regularity and with an optimal control ${|\phi |}_X?g(R)$. Two cases are treated here:(i) ${|?|}_X$ is a $W^{s,p}(0,1)$-seminorm, with $0<s<1<p$ and $sp>1$. We find a lifting φ such that ${|\phi |}_{W^{s,p}(I)}?C(R+R^{1/s})$ and we show that the exponent $1/s$ cannot be improved.(ii) ${|?|}_X$ is the $\mathit{BV}(\Omega )$-seminorm where $\Omega ?{\mathbf{R}}^d$ is a smooth open set. We give a simplified proof of a previous result [J. Dàvila, R. Ignat, Lifting of BV functions with values in $S^1$, C. R. Acad. Sci. Paris, Ser. I 337 (3) (2003) 159–164]: there exists $\phi \in \mathit{BV}(\Omega )$ satisfying ${|\phi |}_{\mathit{BV}}?2R$. To cite this article: B. Merlet, C. R. Acad. Sci. Paris, Ser. I 343 (2006).