arXiv subjects

Mostafa Fazly

Publications and source records attributed to Mostafa Fazly.

30 records · Page 2Linked to original sources

arXiv2015

A pointwise inequality for the fourth order Lane-Emden equation

We prove that the following pointwise inequality holds \begin{equation*} -Δu \ge \sqrt\frac{2}{(p+1)-c_n} |x|^{\frac{a}{2}} u^{\frac{p+1}{2}} + \frac{2}{n-4} \frac{|\nabla u|^2}{u} \ \ \text{in}\ \ \mathbb{R}^n \end{equation*} where $c_n:=\frac{8}{n(n-4)}$, for positive bounded solutions of the fourth order Hénon equation that is \begin{equation*} Δ^2 u = |x|^a u^p \ \ \ \ \text {in }\ \ \mathbb{R}^n \end{equation*} for some $a\ge0$ and $p>1$. Motivated by the Moser's proof of the Harnack's inequality as well as Moser iteration type arguments in the regularity theory, we develop an iteration argument to prove the above pointwise inequality. As far as we know this is the first time that such an argument is applied towards constructing pointwise inequalities for partial differential equations. An interesting point is that the coefficient $\frac{2}{n-4}$ also appears in the fourth order $Q$-curvature and the Paneitz operator. This in particular implies that the scalar curvature of the conformal metric with conformal factor $u^\frac{4}{n-4}$ is positive.

math.AP↗

arXiv2015

One-dimensional symmetry for integral systems in two dimensions

The purpose of this brief paper is to prove De Giorgi type results for stable solutions of the following nonlocal system of integral equations in two dimensions $$ L(u_i) = H_i(u) \quad \text{in} \ \ \mathbb R^2 , $$ where $u=(u_i)_{i=1}^m$ for $u_i: \mathbb R^n\to \mathbb R$, $H=(H_i)_{i=1}^m$ is a general nonlinearity. The operator $L$ is given by $$L(u_i (x)):= \int_{\mathbb R^2} [u_i(x) - u_i(z)] K(z-x) dz,$$ for some kernel $K$. The idea is to apply a linear Liouville theorem for the quotient of partial derivatives, just like in the proof of the classical De Giorgi's conjecture in lower dimensions. Since there is no Caffarelli-Silvestre local extension problem associated to the above operator, we deal with this problem directly via certain integral estimates.

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arXiv2014

Rigidity results for stable solutions of symmetric systems

We study stable solutions of the following nonlinear system $$ -Δu = H(u) \quad \text{in} \ \ Ω$$ where $u:\mathbb R^n\to \mathbb R^m$, $H:\mathbb R^m\to \mathbb R^m$ and $Ω$ is a domain in $\mathbb R^n$. We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of $H$ is symmetric. It seems that this concept is crucial to prove Liouville theorems, when $Ω=\mathbb R^n$, and regularity results, when $Ω=B_1$, for stable solutions of the above system for a general nonlinearity $H \in C^1(\mathbb R ^m)$. Moreover, we provide an improvement for a linear Liouville theorem given in [20] that is a key tool to establish De Giorgi type results in lower dimensions for elliptic equations and systems.

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arXiv2014

Regularity of the extremal solutions associated to elliptic systems

We examine the two elliptic systems given by [(G)_{λ,γ} \quad -Δu = λf'(u) g(v), \quad -Δv = γf(u) g'(v) \quad in $ Ω$,] and [(H)_{λ,γ} \quad -Δu = λf(u) g'(v), \quad -Δv = γf'(u) g(v) \quad in $ Ω$},] with zero Dirichlet boundary conditions and where $ λ,γ$ are positive parameters. We show that for arbitrary nonlinearities $f$ and $g$ that the extremal solutions associated with $ (G)_{λ,γ}$ are bounded provided $ Ω$ is a convex domain in $ \mathbb R^N$ where $ N \le 3$. In the case of a radial domain we show the extremal solutions are bounded provided $ N <10$. The extremal solutions associated with $ (H)_{λ,γ}$ are bounded in the case where $ f$ is arbitrary, $ g(v)=(v+1)^q$ where $ 1 <q<\infty$ and where $ Ω$ is a bounded convex domain in $ \mathbb R^N$, $ N \le 3$. Results are also obtained in higher dimensions for $ (G)_{λ,γ}$ and $(H)_{λ,γ}$ for the case of explicit nonlinearities of the form $ f(u)=(u+1)^p$ and $ g(v)=(v+1)^q$.

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arXiv2013

Higher-dimensional solutions for a nonuniformly elliptic equation

We prove $m$-dimensional symmetry results, that we call $m$-Liouville theorems, for stable and monotone solutions of the following nonuniformly elliptic equation \begin{eqnarray*}\label{mainequ} - div(γ(\mathbf x') \nabla u(\mathbf x)) =λ(\mathbf x' ) f(u(\mathbf x)) \ \ \text{for}\ \ \mathbf x=(\mathbf x',\mathbf x'')\in\mathbf{R}^d\times\mathbf{R}^{s}=\mathbf{R}^n, \end{eqnarray*} where $0\le m<n$ and $0<λ,γ$ are smooth functions and $f\in C^1(\mathbf R)$. The interesting fact is that the decay assumptions on the weight function $γ(\mathbf x') $ play the fundamental role in deriving $m$-Liouville theorems. We show that under certain assumptions on the sign of the nonlinearity $f$, the above equation satisfies a 0-Liouville theorem. More importantly, we prove that for the double-well potential nonlinearities, i.e. $f(u)=u-u^3$, the above equation satisfies a $(d+1)$-Liouville theorem. This can be considered as a higher dimensional counterpart of the celebrated conjecture of De Giorgi for the Allen-Cahn equation. The remarkable phenomenon is that the $\tanh$ function that is the profile of monotone and bounded solutions of the Allen-Cahn equation appears towards constructing higher dimensional Liouville theorems.

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arXiv2013

Liouville theorems for the polyharmonic Henon-Lane-Emden system

We study Liouville theorems for the following polyharmonic Hénon-Lane-Emden system \begin{eqnarray*} \left\{\begin{array}{lcl} (-Δ)^m u&=& |x|^{a}v^p \ \ \text{in}\ \ \mathbb{R}^n,\\ (-Δ)^m v&=& |x|^{b}u^q \ \ \text{in}\ \ \mathbb{R}^n, \end{array}\right. \end{eqnarray*} when $m,p,q \ge 1,$ $pq\neq1$, $a,b\ge0$. The main conjecture states that $(u,v)=(0,0)$ is the unique nonnegative solution of this system whenever $(p,q)$ is {\it under} the critical Sobolev hyperbola, i.e. $ \frac{n+a}{p+1}+\frac{n+b}{q+1}>{n-2m}$. We show that this is indeed the case in dimension $n=2m+1$ for bounded solutions. In particular, when $a=b$ and $p=q$, this means that $u=0$ is the only nonnegative bounded solution of the polyharmonic Hénon equation \begin{equation*} (-Δ)^m u= |x|^{a}u^p \ \ \text{in}\ \ \mathbb{R}^{n} \end{equation*} in dimension $n=2m+1$ provided $p$ is the subcritical Sobolev exponent, i.e., $1<p<{1+4m+2a}$. Moreover, we show that the conjecture holds for radial solutions in any dimensions. It seems the power weight functions $|x|^a$ and $|x|^b$ make the problem dramatically more challenging when dealing with nonradial solutions.

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arXiv2013

Effect of weights on stable solutions of a quasilinear elliptic equation

In this note, we study Liouville theorems for the stable and finite Morse index weak solutions of the quasilinear elliptic equation $-Δ_p u= f(x) F(u) $ in $\mathbb{R}^n$ where $p\ge 2$, $0\le f\in C(\mathbb{R}^n)$ and $F\in C^1(\mathbb{R})$. We refer to $f(x)$ as {\it weight} and to $F(u)$ as {\it nonlinearity}. The remarkable fact is that if the weight function is bounded from below by a strict positive constant that is $0 p-1$ and $-u^{q}$ where $q<0$, known as the Gelfand, the Lane-Emden and the negative exponent nonlinearities, respectively, we prove Liouville theorems for both radial finite Morse index (not necessarily bounded) and stable (not necessarily radial nor bounded) solutions.

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arXiv2012

On the Hénon-Lane-Emden conjecture

We consider Liouville-type theorems for the following Hénon-Lane-Emden system \hfill -Δu&=& |x|^{a}v^p \text{in} \mathbb{R}^N, \hfill -Δv&=& |x|^{b}u^q \text{in} \mathbb{R}^N, when $pq>1$, $p,q,a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $ \frac{N+a}{p+1}+\frac{N+b}{q+1}>{N-2}$. We show that this is indeed the case in dimension N=3 provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case. Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $N\ge 3$ in the first case (resp., $N\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $ 1<p<\frac{N+2+2a}{N-2}$ (resp., $ 1<p<\frac{N+4+2a}{N-4}$). Finally, we show that non-negative stable solutions of the full Hénon-Lane-Emden system are trivial provided \label{sysdim00} N<2+2(\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}}+ \sqrt{\frac{pq(q+1)}{p+1}-\sqrt\frac{pq(q+1)}{p+1}}).

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arXiv2012

Uniqueness of solutions for a nonlocal elliptic eigenvalue problem

We examine equations of the form {eqnarray*} \{{array}{lcl} \hfill \HA u &=& λg(x) f(u) \qquad \text{in}\ Ω\hfill u&=& 0 \qquad \qquad \qquad \text{on}\ \pOm, {array}. {eqnarray*} where $ λ>0$ is a parameter and $ Ω$ is a smooth bounded domain in $ \IR^N$, $ N \ge 2$. Here $ g$ is a positive function and $ f$ is an increasing, convex function with $ f(0)=1$ and either $ f$ blows up at 1 or $ f$ is superlinear at infinity. We show that the extremal solution $u^*$ associated with the extremal parameter $ λ^*$ is the unique solution. We also show that when $f$ is suitably supercritical and $ Ω$ satisfies certain geometrical conditions then there is a unique solution for small positive $ λ$.

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arXiv2012

De Giorgi type results for elliptic systems

We consider the following elliptic system Δu =\nabla H (u) \ \ \text{in}\ \ \mathbf{R}^N, where $u:\mathbf{R}^N\to \mathbf{R}^m$ and $H\in C^2(\mathbf{R}^m)$, and prove, under various conditions on the nonlinearity $H$ that, at least in low dimensions, a solution $u=(u_i)_{i=1}^m$ is necessarily one-dimensional whenever each one of its components $u_i$ is monotone in one direction. Just like in the proofs of the classical De Giorgi's conjecture in dimension 2 (Ghoussoub-Gui) and in dimension 3 (Ambrosio-Cabré), the key step is a Liouville theorem for linear systems. We also give an extension of a geometric Poincaré inequality to systems and use it to establish De Giorgi type results for stable solutions as well as additional rigidity properties stating that the gradients of the various components of the solutions must be parallel. We introduce and exploit the concept of {\it an orientable system}, which seems to be key for dealing with systems of three or more equations. For such systems, the notion of a stable solution in a variational sense coincide with the pointwise (or spectral) concept of stability.

math.AP↗

arXiv2011

Liouville type theorems for stable solutions of certain elliptic systems

We establish Liouville type theorems for elliptic systems with various classes of non-linearities on $\mathbb{R}^N$. We show among other things, that a system has no semi-stable solution in any dimension, whenever the infimum of the derivatives of the corresponding non-linearities is positive. We give some immediate applications to various standard systems, such as the Gelfand, and certain Hamiltonian systems. The case where the infimum is zero is more interesting and quite challenging. We show that any $C^2(\mathbb{R}^N)$ positive entire semi-stable solution of the following Lane-Emden system, {eqnarray*} \hbox{$(N_{λ,γ})$}50pt \{{array}{lcl} \hfill -Δu&=&λf(x) \ v^p, \hfill -Δv&=&γf(x) \ u^q, {array}.{eqnarray*} is necessarily constant, whenever the dimension $N< 8+3α+\frac{8+4α}{q-1}$, provided $p=1$, $q\ge2$ and $f(x)= (1+|x|^2)^{\fracα{2}} $. The same also holds for $p=q\ge2$ provided $N < 2+ \frac{2(2+α)}{p-1} (p+\sqrt{p(p-1)})$. We also consider the case of bounded domains $Ω\subset\mathbb{R}^N$, where we extend results of Brown et al. \cite{bs} and Tertikas \cite{te} about stable solutions of equations to systems. At the end, we prove a Pohozaev type theorem for certain weighted elliptic systems.

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arXiv2011

On stable entire solutions of semi-linear elliptic equations with weights

We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of {equation} \label{pop} -div(ω_1 \nabla u) = ω_2 f(u) \qquad \text{in}\ \ \IR^N, {equation} with positive smooth weights $ ω_1(x),ω_2(x)$. We consider the cases $ f(u) = e^u, u^p$ where $p>1$ and $ -u^{-p}$ where $ p>0$. We obtain various non-existence results which depend on the dimension $N$ and also on $ p$ and the behaviour of $ ω_1,ω_2$ near infinity. Also the monotonicity of $ ω_1$ is involved in some results. Our methods here are the methods developed by Farina, \cite{f2}. We examine a specific class of weights $ ω_1(x) = (|x|^2 +1)^\fracα{2}$ and $ ω_2(x) = (|x|^2+1)^\fracβ{2} g(x)$ where $ g(x)$ is a positive function with a finite limit at $ \infty$. For this class of weights non-existence results are optimal. To show the optimality we use various generalized Hardy inequalities.

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