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Mostafa Fazly

Publications and source records attributed to Mostafa Fazly.

At least 19 recordsLinked to original sources
arXiv2020

On stable and finite Morse index solutions of the nonlocal H\'{e}non-Gelfand-Liouville equation

We consider the nonlocal H\'{e}non-Gelfand-Liouville problem $$ (-\Delta)^s u = |x|^a e^u\quad\mathrm{in}\quad \mathbb R^n, $$ for every $s\in(0,1)$, $a>0$ and $n>2s$. We prove a monotonicity formula for solutions of the above equation using rescaling arguments. We apply this formula together with blow-down analysis arguments and technical integral estimates to establish non-existence of finite Morse index solutions when $$\dfrac{\Gamma(\frac n2)\Gamma(s)}{\Gamma(\frac{n-2s}{2})}\left(s+\frac a2\right)> \dfrac{\Gamma^2(\frac{n+2s}{4})}{\Gamma^2(\frac{n-2s}{4})}.$$

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arXiv2020

Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations

We study the quasilinear elliptic equation \begin{equation*} -Qu=e^u \ \ \text{in} \ \ \Omega\subset \mathbb{R}^{N} \end{equation*} where the operator $Q$, known as Finsler-Laplacian (or anisotropic Laplacian), is defined by $$Qu:=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)),$$ where $F_{\xi_{i}}=\frac{\partial F}{\partial\xi_{i}}$ and $F: \mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of $ C^{2}(\mathbb{R}^{N}\setminus\{0\})$, that satisfies certain assumptions. For bounded domain $\Omega$ and for a stable weak solution of the above equation, we prove that the Hausdorff dimension of singular set does not exceed $N-10$. For the entire space, we apply Moser iteration arguments, established by Dancer-Farina and Crandall-Rabinowitz in the context, to prove Liouville theorems for stable solutions and for finite Morse index solutions in dimensions $N<10$ and $2<N<10$, respectively. We also provide an explicit solution that is stable outside a compact set in $N=2$. In addition, we provide similar Liouville theorems for the power-type nonlinearities.

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arXiv2020

On stable and finite Morse index solutions of the fractional Toda system

We develop a monotonicity formula for solutions of the fractional Toda system $$ (-\Delta)^s f_\alpha = e^{-(f_{\alpha+1}-f_\alpha)} - e^{-(f_\alpha-f_{\alpha-1})} \quad \text{in} \ \ \mathbb R^n,$$ when $0 2s$ and $$ \dfrac{\Gamma(\frac{n}{2})\Gamma(1+s)}{\Gamma(\frac{n-2s}{2})} \frac{Q(Q-1)}{2} > \frac{ \Gamma(\frac{n+2s}{4})^2 }{ \Gamma(\frac{n-2s}{4})^2} . $$ Here, $\Gamma$ is the Gamma function. When $Q=2$, the above equation is the classical (fractional) Gelfand-Liouville equation.

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arXiv2020

Classification of finite Morse index solutions of higher-order Gelfand-Liouville equation

We classify finite Morse index solutions of the following Gelfand-Liouville equation \begin{equation*} (-\Delta)^{s} u= e^u \ \ \text{in} \ \ \mathbb{R}^n, \end{equation*} for $1 2s$ and \begin{equation*} \label{1.condition} \frac{ \Gamma(\frac{n+2s}{4})^2 }{ \Gamma(\frac{n-2s}{4})^2} < \frac{\Gamma(\frac{n}{2}) \Gamma(1+s)}{ \Gamma(\frac{n-2s}{2})}, \end{equation*} where $\Gamma$ is the classical Gamma function. The cases of $s=1$ and $s=2$ are settled by Dancer and Farina \cite{df,d} and Dupaigne et al. \cite{dggw}, respectively, using Moser iteration arguments established by Crandall and Rabinowitz \cite{CR}. The case of $0<s<1$ is established by Hyder-Yang in \cite{hy} applying arguments provided in \cite{ddw,fw}.

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arXiv2019

Analysis of propagation for impulsive reaction-diffusion models

We study a hybrid impulsive reaction-advection-diffusion model given by a reaction-advection-diffusion equation composed with a discrete-time map in space dimension $n\in\mathbb N$. The reaction-advection-diffusion equation takes the form \begin{equation*}\label{} u^{(m)}_t = \text{div}(A\nabla u^{(m)}-q u^{(m)}) + f(u^{(m)}) \quad \text{for} \ \ (x,t)\in\mathbb R^n \times (0,1] , \end{equation*} for some function $f$, a drift $q$ and a diffusion matrix $A$. When the discrete-time map is local in space we use $N_m(x)$ to denote the density of population at a point $x$ at the beginning of reproductive season in the $m$th year and when the map is nonlocal we use $u_m(x)$. The local discrete-time map is \begin{eqnarray*}\label{}\left\{ \begin{array}{lcl} u^{(m)}(x,0) = g(N_m(x)) \quad \text{for} \ \ x\in \mathbb R^n , \\ N_{m+1}(x):=u^{(m)}(x,1) \quad \text{for} \ \ x\in \mathbb R^n , \end{array}\right. \end{eqnarray*} for some function $g$. The nonlocal discrete time map is \begin{eqnarray*}\label{}\left\{ \begin{array}{lcl} u^{(m)}(x,0) = u_{m}(x) \quad \text{for} \ \ x\in \mathbb R^n , \\ \label{mainb2} u_{m+1}(x) := g\left(\int_{\mathbb R^n} K(x-y)u^{(m)}(y,1) dy\right) \quad \text{for} \ \ x\in \mathbb R^n, \end{array}\right. \end{eqnarray*} when $K$ is a nonnegative normalized kernel. SEE THE ARTICLE FOR COMPLETE ABSTRACT.

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arXiv2019

On nonlocal systems with jump processes of finite range and with decays

We study the following system of equations $$ L_i(u_i) = H_i(u_1,\cdots,u_m) \quad \text{in} \ \ \mathbb R^n , $$ when $m\ge 1$, $u_i: \mathbb R^n \to \mathbb R$ and $H=(H_i)_{i=1}^m$ is a sequence of general nonlinearities. The nonlocal operator $L_i$ is given by $$L_i(f (x)):= \lim_{ε\to 0} \int_{\mathbb R^n \setminus B_ε(x) } [f(x) - f(z)] J_i(z-x) dz,$$ for a sequence of even, nonnegative and measurable jump kernels $J_i$. We prove a Poincaré inequality for stable solutions of the above system for a general jump kernel $J_i$. In particular, for the case of scalar equations, that is when $m=1$, it reads \begin{equation*}\label{} \iint_{ \mathbb R^{2n}} \mathcal A_y(\nabla_x u) [η^2(x)+η^2(x+y)] J(y) dx dy \le \iint_{ \mathbb R^{2n}} \mathcal B_y(\nabla_x u) [ η(x) - η(x+y) ] ^2 J(y) d x dy , \end{equation*} for any $η\in C_c^1(\mathbb R^{n})$ and for some nonnegative $ \mathcal A_y(\nabla_x u)$ and $ \mathcal B_y(\nabla_x u)$. This is a counterpart of the celebrated inequality derived by Sternberg and Zumbrun in \cite{sz} for semilinear elliptic equations that is used extensively in the literature to establish De Giorgi type results, to study phase transitions and to prove regularity properties. We then apply this inequality to finite range jump processes and to jump processes with decays to prove De Giorgi type results in two dimensions. In addition, we show that whenever $H_i(u)\ge 0$ or $\sum_{i=1}^m u_i H_i(u)\le 0$ then Liouville theorems hold for each $u_i$ in one and two dimensions. Lastly, we provide certain energy estimates under various assumptions on the jump kernel $J_i$ and a Liouville theorem for the quotient of partial derivatives of $u$.

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arXiv2019

Regularity of extremal solutions of nonlocal elliptic systems

We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem \begin{eqnarray} \left\{ \begin{array}{lcl} \hfill \mathcal L u &=& λF(u,v) \qquad \text{in} \ \ Ω, \\ \hfill \mathcal L v &=& γG(u,v) \qquad \text{in} \ \ Ω, \\ \hfill u,v &=&0 \qquad \qquad \text{on} \ \ \mathbb R^n\setminusΩ, \end{array}\right. \end{eqnarray} with an integro-differential operator, including the fractional Laplacian, of the form \begin{equation*}\label{} \mathcal L(u (x))= \lim_{ε\to 0} \int_{\mathbb R^n\setminus B_ε(x) } [u(x) - u(z)] J(z-x) dz , \end{equation*} when $J$ is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of $J(y)=\frac{a(y/|y|)}{|y|^{n+2s}}$ where $s\in (0,1)$ and $a$ is any nonnegative even measurable function in $L^1(\mathbb {S}^{n-1})$ that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions $n < 10s$ and $ n<2s+\frac{4s}{p\mp 1}[p+\sqrt{p(p\mp1)}]$ for the Gelfand and Lane-Emden systems when $p>1$ (with positive and negative exponents), respectively. When $s\to 1$, these dimensions are optimal. However, for the case of $s\in(0,1)$ getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions $n<4s$. As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.

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arXiv2019

De Giorgi type results for equations with nonlocal lower-order terms

It is known that the De Giorgi's conjecture does not hold in two dimensions for semilinear elliptic equations with a nonzero drift, in general, $$ Δu+ q\cdot \nabla u+f(u)=0 \ \ \text{in } \ \ \mathbb R^2, $$ when $q=(0,-c)$ for $c\neq 0$. This equation arises in the modeling of Bunsen burner flames. Bunsen flames are usually made of two flames: a diffusion flame and a premixed flame. In this article, we prove De Giorgi type results, and stability conjecture, for the following local-nonlocal counterpart of the above equation (with a nonlocal premixed flame) in two dimensions, $$Δu + c L[u] + f(u)=0 \quad \text{in} \ \ \mathbb R^n, $$ when $L$ is a nonlocal operator, $f\in C^1(\mathbb R)$ and $c\in\mathbb R^+$. In addition, we provide a priori estimates for the above equation, when $n\ge 1$, with various jumping kernels. The operator $Δ+cL$ is an infinitesimal generator of jump-diffusion processes in the context of probability theory.

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arXiv2019

Monotonicity formulas for coupled elliptic gradient systems with applications

Consider the following coupled elliptic system of equations \begin{equation*} \label{} (-Δ)^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} \ \ \mathbb{R}^n , \end{equation*} where $0 1$, $m\ge1$, $u=(u_i)_{i=1}^m$ and $u_i:\mathbb R^n\to \mathbb R$. The qualitative behavior of solutions of the above system has been studied from various perspectives in the literature including the free boundary problems and the classification of solutions. For the case of local scalar equation, that is when $m=1$ and $s=1$, Gidas and Spruck in \cite{gs} and later Caffarelli, Gidas and Spruck in \cite{cgs} provided the classification of solutions for Sobolev sub-critical and critical exponents. More recently, for the case of local system of equations that is when $m\ge1$ and $s=1$ a similar classification result is given by Druet, Hebey and Vétois in \cite{dhv} and references therein. In this paper, we derive monotonicity formulae for entire solutions of the above local, when $s=1,2$, and nonlocal, when $0<s<1$ and $1<s<2$, system. These monotonicity formulae are of great interests due to the fact that a counterpart of the celebrated monotonicity formula of Alt-Caffarelli-Friedman \cite{acf} seems to be challenging to derive for system of equations. Then, we apply these formulae to give a classification of finite Morse index solutions. In the end, we provide an open problem in regards to monotonicity formulae for Lane-Emden systems.

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arXiv2018

Symmetry properties for solutions of nonlocal equations involving nonlinear operators

We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional $p-$Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.

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arXiv2016

On impulsive reaction-diffusion models in higher dimensions

Assume that $N_m(x)$ denotes the density of the population at a point $x$ at the beginning of the reproductive season in the $m$th year. We study the following impulsive reaction-diffusion model for any $m\in \mathbb Z^+$ \begin{eqnarray*}\label{} \ \ \ \ \ \left\{ \begin{array}{lcl} u^{(m)}_t = div(A\nabla u^{(m)}-a u^{(m)}) + f(u^{(m)}) \quad \text{for} \ \ (x,t)\inΩ\times (0,1] u^{(m)}(x,0)=g(N_m(x)) \quad \text{for} \ \ x\in Ω N_{m+1}(x):=u^{(m)}(x,1) \quad \text{for} \ \ x\in Ω\end{array}\right. \end{eqnarray*} for functions $f,g$, a drift $a$ and a diffusion matrix $A$ and $Ω\subset \mathbb R^n$. Study of this model requires a simultaneous analysis of the differential equation and the recurrence relation. When boundary conditions are hostile we provide critical domain results showing how extinction versus persistence of the species arises, depending on the size and geometry of the domain. We show that there exists an {\it extreme volume size} such that if $|Ω|$ falls below this size the species is driven extinct, regardless of the geometry of the domain. To construct such extreme volume sizes and critical domain sizes, we apply Schwarz symmetrization rearrangement arguments, the classical Rayleigh-Faber-Krahn inequality and the spectrum of uniformly elliptic operators. The critical domain results provide qualitative insight regarding long-term dynamics for the model. Lastly, we provide applications of our main results to certain biological reaction-diffusion models regarding marine reserve, terrestrial reserve, insect pest outbreak and population subject to climate change.

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arXiv2016

Solutions of multi-component fractional symmetric systems

We study the following elliptic system concerning the fractional Laplacian operator $$(- Δ)^ {s_i} u_i = H_i ( u_1,\cdots,u_m) \ \ \text{in}\ \ \mathbb{R}^n,$$ when $0 \max(0,1-2\min \left \{s_i \right \})$ for $1\le i \le m$. The above system is called symmetric when the matrix $\mathcal H=(\partial_j H_i(u_1,\cdots,u_m))_{i,j=1}^m$ is symmetric. The notion of symmetric systems seems crucial to study this system with a general nonlinearity $H=(H_i)_{i=1}^m$. We establish De Giorgi type results for stable and $H$-monotone solutions of symmetric systems in lower dimensions that is either $n=2$ and $0<s_i<1$ or $n=3$ and $1/2 \le \min\{s_i\}<1$. The case that $n=3$ and at least one of parameters $s_i$ belongs to $(0,1/2)$ remains open as well as the case $n \ge 4$. Applying a geometric Poincaré inequality, we conclude that gradients of components of solutions are parallel in lower dimensions when the system is coupled. More precisely, we show that the angle between vectors $\nabla u_i$ and $\nabla u_j$ is exactly $\arccos\left({|\partial_j H_i(u)|}/{\partial_j H_i(u)}\right)$. In addition, we provide Hamiltonian identities, monotonicity formulae and Liouville theorems. Lastly, we apply some of our main results to a two-component nonlinear Schrödinger system, that is a particular case of the above system, and we prove Liouville theorems and monotonicity formulae.

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arXiv2016

Stable solutions of symmetric systems on Riemannian manifolds

We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold $\mathbb{M}$ without boundary, \begin{equation*} -Δ_g u_i = H_i(u_1,\cdots,u_m) \ \ \text{on} \ \ \mathbb{M}, \end{equation*} when $Δ_g$ stands for the Laplace-Beltrami operator, $u_i:\mathbb{M}\to \mathbb R$ and $H_i\in C^1(\mathbb R^m) $ for $1\le i\le m$. This system is called symmetric if the matrix of partial derivatives of all components of $H$, that is $\mathbb H(u)=(\partial_j H_i(u))_{i,j=1}^m$, is symmetric. We prove a stability inequality and a Poincaré type inequality for stable solutions using the Bochner-Weitzenböck formula. Then, we apply these inequalities to establish Liouville theorems and flatness of level sets for stable solutions of the above symmetric system, under certain assumptions on the manifold and on solutions.

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arXiv2015

Symmetry results for fractional elliptic systems and related problems

We study elliptic gradient systems with fractional laplacian operators on the whole space $$ (- Δ)^\mathbf s \mathbf u =\nabla H (\mathbf u) \ \ \text{in}\ \ \mathbf{R}^n,$$ where $\mathbf u:\mathbf{R}^n\to \mathbf{R}^m$, $H\in C^{2,γ}(\mathbf{R}^m)$ for $γ> \max(0,1-2\min \left \{s_i \right \})$, $\mathbf s=(s_1,\cdots,s_m)$ for $0<s_i<1$ and $\nabla H (\mathbf u)=(H_{u_i}(u_1, u_2,\cdots,u_m))_{i}$. We prove De Giorgi type results for this system for certain values of $\mathbf s$ and in lower dimensions, i.e. $n=2,3$. Just like the local case, the concepts of orientable systems and $H-$monotone solutions, established in [18], play the key role in proving symmetry results. In addition, we provide optimal energy estimates, a monotonicity formula, a Hamiltonian identity and various Liouville theorems.

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arXiv2015

On stable solutions of the fractional Henon-Lane-Emden equation

We derive monotonicity formulae for solutions of the fractional Hénon-Lane-Emden equation \begin{equation*} (-Δ)^{s} u=|x|^a |u|^{p-1} u \ \ \ \text{in } \ \ \mathbb{R}^n, \end{equation*} when $0 0$ and $p>1$. Then, we apply these formulae to classify stable solutions of the above equation.

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arXiv2015

Entire solutions of quasilinear symmetric systems

We study the following quasilinear elliptic system for all $i=1,\cdots,m$ \begin{equation*} \label{} -div(Φ'(|\nabla u_i|^2) \nabla u_i) = H_i(u) \quad \text{in} \ \ \mathbb{R}^n \end{equation*} where $u=(u_i)_{i=1}^m: \mathbb R^n\to \mathbb R^m$ and the nonlinearity $ H_i(u) \in C^1(\mathbb R^m)\to \mathbb R$ is a general nonlinearity. Several celebrated operators such as the prescribed mean curvature, the Laplacian and the $p$-Laplacian operators fit in the above form, for appropriate $Φ$. We establish a Hamiltonian identity of the following form for all $x_n\in\mathbb R$ \begin{equation*}\label{} \int_{\mathbb R^{n-1}} \left(\sum_{i=1}^{m} \left[ \frac{1}{2} Φ\left(|\nabla u_i|^2\right) - Φ'\left(|\nabla u_i|^2\right) |\partial_{x_n} u_i|^2 \right] - \tilde H(u) \right) d x'\equiv C, \end{equation*} where $x=(x',x_n)\in\mathbb R^{n}$ and $\tilde H$ is the antiderivative of $H=(H_i)_{i=1}^m$. This can be seen as a counterpart of celebrated pointwise inequalities provided by Caffarelli, Garofalo and Segala in \cite{cgs} and by Modica in \cite{m}. For the case of system of equations, that is when $m\ge 2$, we show that as long as $$α\ge α^*:=\inf_{s>0}\left\{\frac{2 s Φ'(s)}{Φ(s)}\right\}$$ the function $I_α(r):=\frac{1}{r^{n-α}} \int_{B_r} \sum_{i=1}^{m} Φ(|\nabla u_i|^2) - 2\tilde H(u)$ is monotone nondecreasing in $r$. We call this a weak monotonicity formula since for $m=1$ it is shown in \cite{cgs} that $I_α(r)$ is monotone when $α\ge 1$, under certain conditions on $Φ$. We prove De Giorgi type results and Liouville theorems for $H$-monotone and stable solutions in two and three dimensions when the system is symmetric.

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arXiv2015

On finite Morse index solutions of higher order fractional Lane-Emden equations

We classify finite Morse index solutions of the fractional Lane-Emden equation $(-Δ)^{s} u=|u|^{p-1} u \ \ \ \mathbb{R}^n $ for $1<s<2$. For the local case, $s=1$ and $s=2$ this classification was done by Farina in [10] and Davila, Dupaigne, Wang and Wei in [8], respectively. Moreover, for the nonlocal case, $0<s<1$, finite Morse index solutions are classified by Davila, Dupaigne and Wei in [7].

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