arXiv subjects

A. Aghajani

Publications and source records attributed to A. Aghajani.

6 recordsLinked to original sources

arXiv2021

A nonlinear elliptic problem involving the gradient on a half space

We consider perturbations of the diffusive Hamilton-Jacobi equation \begin{equation*} %\label{non_pert} \left\{ \begin{array}{lcl} \hfill -Δu &=& (1+g(x))| \nabla u|^p\qquad \mbox{ in } \IR^N_+, \\ \hfill u &=& 0 \hfill \mbox{ on } \partial \IR^N_+, \end{array}\right. \end{equation*} for $ p>1$. We prove the existence of a classical solution provided $ p \in (\frac{4}{3},2)$ and $g$ is bounded with uniform radial decay to zero.

math.AP↗

arXiv2020

Some elliptic problems involving the gradient on general bounded and exterior domains

In this article we consider the existence of positive singular solutions on bounded domains and also classical solutions on exterior domains. First we consider positive singular solutions of the following problems: \begin{equation} \label{eq_abst_1}-Δu = (1+g(x)) | \nabla u|^p \qquad \mbox{ in } B_1, \qquad u = 0 \mbox{ on } \;\; \partial B_1, \qquad \mbox{ and} \end{equation} \begin{equation} \label{eq_abst_2} -Δu = | \nabla u|^p \qquad \mbox{ in } Ω, \qquad u = 0 \mbox{ on } \;\; \partial Ω. \end{equation} In the first problem $B_1$ is the unit ball in $ \mathbb{R}^N$ and in the second $Ω$ is a bounded smooth domain in $ \mathbb{R}^N$. In both cases we assume $ N \ge 3$, $ \frac{N}{N-1} \frac{N}{N-1}$. We prove the existence of a bounded positive classical solution with the additional property that $ \nabla u(x) \cdot x>0$ for large $|x|$.

math.AP↗

arXiv2018

A note on the nonexistence of positive supersolutions to elliptic equations with gradient terms

We prove that if the elliptic problem $-Δu+b(x)|\nabla u|=c(x)u$ with $c\ge0$ has a positive supersolution in a domain $Ω$ of $ \IR^{N\ge 3}$, then $c,b$ must satisfy the inequality \[\sqrt{ \int_Ωcϕ^2}\le \sqrt{ \int_Ω| \nablaϕ|^2}+\sqrt{ \int_Ω\frac{b^2}{4}ϕ^2},~~~ϕ\in C_c^\infty(Ω).\] As an application, we obtain Liouville type theorems for positive supersolutions in exterior domains when $c(x)-\frac{b^2(x)}{4}>0$ for large $|x|$, but unlike the known results we allow the case $\liminf_{|x|\rightarrow\infty}c(x)-\frac{b^2(x)}{4}=0$. Also the weights $b$ and $c$ are allowed to be unbounded. In particular, among other things, we show that if $τ:=\limsup_{|x| \rightarrow\infty}|xb(x)|<\infty$ then this problem does not admit any positive supersolution if \[\liminf_{|x| \rightarrow\infty}|x|^2c(x)> \frac{(N-2+τ)^2}{4},\] and, when $τ=\infty, $ we have the same if \[\limsup_{R\rightarrow\infty} R\Big(\frac{ \inf_{R<|x|<2 R} (c(x)-\frac{b(x)^2}{4})}{\sup_{\frac{R}{2}<|x|<4 R}|b(x)|}\Big)=\infty.\]

math.AP↗

arXiv2018

Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications

In this paper we consider positive supersolutions of the nonlinear elliptic equation \[- Δu = ρ(x) f(u)|\nabla u|^p, \qquad \hfill \mbox{ in } Ω,\] where $0\le p<1$, $ Ω$ is an arbitrary domain (bounded or unbounded) in $ \IR^N$ ($N\ge 2$), $f: [0,a_{f}) \rightarrow \Bbb{R}_{+}$ $(0 < a_{f} \leqslant +\infty)$ is a non-decreasing continuous function and $ρ: Ω\rightarrow \IR$ is a positive function. Using the maximum principle we give explicit estimates on positive supersolutions $u$ at each point $x\inΩ$ where $\nabla u\not\equiv0$ in a neighborhood of $x$. As consequences, we discuss the dead core set of supersolutions on bounded domains, and also obtain Liouville type results in unbounded domains $Ω$ with the property that $\sup_{x\inΩ}dist (x,\partialΩ)=\infty$.

math.AP↗

arXiv2017

Regularity of the extremal solutions associated to elliptic systems

We examine the elliptic system given by \begin{eqnarray*} \qquad \left\{ \begin{array}{lcl} -Δu =λf(v) \quad \mbox{ in } Ω -Δv =γf(u) \quad \mbox{ in } Ω, u=v =0, \quad \mbox{ on } \pOm \end{array}\right. \end{eqnarray*} where $λ,γ$ are positive parameters, $Ω$ is a smooth bounded domain in $\IR^N$ and $f$ is a $C^{2}$ positive, nondecreasing and convex function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. Assuming $$0<τ_{-}:=\liminf_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}}\leq τ_{+}:=\limsup_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}}\leq 2,$$ we show that the extremal solution $(u^*, v^*)$ associated to the above system is smooth provided\\ $N<\frac{2α_{*}(2-τ_{+})+2τ_{+}}{τ_{+}}\max\{1,τ_{+}\}$, where $α_{*}>1$ denotes the largest root of the $2^{nd}$ order polynomial $$P_{f}(α,τ_{-},τ_{+}):=(2-τ_{-})^{2} α^{2}- 4(2-τ_{+})α+4(1-τ_{+}).$$ As a consequences, $u^*, v^*\in L^\infty(Ω)$ for $N<5$. Moreover, if $τ_{-}=τ_{+}$, then $u^*, v^*\in L^\infty(Ω)$ for $N<10$.

math.AP↗

arXiv2016

Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities

We consider the fourth order problem $Δ^{2}u=λf(u)$ on a general bounded domain $Ω$ in $R^{n}$ with the Navier boundary condition $u=Δu=0$ on $\partial Ω$. Here, $λ$ is a positive parameter and $ f:[0,a_{f}) \rightarrow \Bbb{R}_{+} $ $ (0 < a_{f} \leqslant \infty)$ is a smooth, increasing, convex nonlinearity such that $ f(0) > 0 $ and which blows up at $ a_{f} $. Let $$0<τ_{-}:=\liminf_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}\leq τ_{+}:=\limsup_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}<2.$$ We show that if $u_{m}$ is a sequence of semistable solutions correspond to $λ_{m}$ satisfy the stability inequality $$ \sqrt{λ_{m}}\int_Ω\sqrt{f'(u_{m})}ϕ^{2}dx\leq \int_Ω|\nablaϕ|^{2}dx, ~~\text{for all}~ϕ\in H^{1}_{0}(Ω),$$ then $\sup_{m} ||u_{m}||_{L^{\infty}(Ω)}<a_{f}$ for $n< \frac{4α_{*}(2-τ_{+})+2τ_{+}}{τ_{+}}\max \{1, τ_{+}\},$ where $α^{*}$ is the largest root of the equation $$(2-τ_{-})^{2} α^{4}- 8(2-τ_{+})α^{2}+4(4-3τ_{+})α-4(1-τ_{+})=0.$$ In particular, if $τ_{-}=τ_{+}:=τ$, then $\sup_{m} ||u_{m}||_{L^{\infty}(Ω)}<a_{f}$ for $n\leq12$ when $τ\leq 1$, and for $n\leq7$ when $τ\leq 1.57863$. These estimates lead to the regularity of the corresponding extremal solution $u^{*}(x)=\lim_{λ\uparrowλ^{*}}u_λ(x),$ where $λ^*$ is the extremal parameter of the eigenvalue problem.

math.AP↗