arXiv subjects

Shaya Shakerian

Publications and source records attributed to Shaya Shakerian.

7 recordsLinked to original sources

arXiv2020

Sharp pointwise estimates for weighted critical $p$-Laplace equations

We investigate the asymptotic behavior of solutions to a class of weighted quasilinear elliptic equations which arise from the Euler--Lagrange equation associated with the Caffarelli--Kohn--Nirenberg inequality. We obtain sharp pointwise estimates which extend and improve previous results obtained in the unweighted case. In particular, we show that we can refine the asymptotic expansion at infinity by using a Kelvin-type transformation, which reduces the problem to another elliptic-type problem near the origin. The application of this transformation is straightforward in the linear case but more delicate in the quasilinear case. In particular, it is necessary in this case to establish some preliminary estimates before being able to apply the transformation.

math.AP↗

arXiv2018

Mass and Extremals Associated with the Hardy-Schrödinger Operator on Hyperbolic Space

We consider the Hardy-Schrödinger operator $ -Δ_{\mathbb{B}^n}-γ{V_2}$ on the Poincaré ball model of the Hyperbolic space ${\mathbb{B}^n}$ ($n \geq 3$). Here $V_2$ is a well chosen radially symmetric potential, which behaves like the Hardy potential around its singularity at $0$, i.e., $V_2(r)\sim \frac{1}{r^2}$. Just like in the Euclidean setting, the operator $ -Δ_{\mathbb{B}^n}-γ{V_2}$ is positive definite whenever $γ<\frac{(n-2)^2}{4}$, in which case we exhibit explicit solutions for the equation $$-Δ_{\mathbb{B}^n}u-γ{V_2}u=V_{2^*(s)}u^{2^*(s)-1}\quad{\text{ in }}\mathbb{B}^n,$$ where $0\leq s <2$, $2^*(s)=\frac{2(n-s)}{n-2}$, and $V_{2^*(s)}$ is a weight that behaves like $\frac{1}{r^s}$ around $0$. The same equation, on bounded domains $Ω$ of ${\mathbb{B}^n}$ containing $0$ but not touching the hyperbolic boundary, has positive solutions if $0 < γ\leq \frac{(n-2)^{2}}{4}-\frac{1}{4}$. However, if $\frac{(n-2)^{2}}{4}-\frac{1}{4}< γ< \frac{(n-2)^{2}}{4}$, the existence of solutions requires the positivity of the "hyperbolic Hardy mass" $m_{_{\mathbb{B}^n}}(Ω)$ of the domain, a notion that we introduce and analyse therein.

math.AP↗

arXiv2017

Existence Result for Non-linearly Perturbed Hardy-Schrödinger Problems: Local and Non-local cases

Let $Ω\subset \mathbb{R}^n$ be a smooth bounded domain having zero in its interior $0 \in Ω.$ We fix $0 < α\le 2$ and $0 \le s <α.$ We investigate a sufficient condition for the existence of a positive solution for the following perturbed problem associated with the Hardy-Schrödinger operator $ L_{γ,α,}: = ({-}{ Δ})^{\fracα{2}}- \fracγ{|x|^α}$ on $Ω:$ \begin{equation*} \left\{\begin{array}{rl} \displaystyle ({-}{ Δ})^{\fracα{2}}u- γ\frac{u}{|x|^α} - λu= {\frac{u^{2_α^*(s)-1}}{|x|^s}}+ h(x) u^{q-1} & \text{in } Ω\\ u=0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & \text{in } \mathbb{R}^n \setminus Ω, \end{array}\right. \end{equation*} where ${2_α^*(s)}:=\frac{2(n-s)}{n-α},$ $λ\in \mathbb{R} $, $h \in C^0(\overlineΩ),$ $h \ge 0,$ $q \in (2, 2^*_α)$ with $2^*_α:=2^*_α(0),$ and $γ< γ_H(α),$ the latter being the best constant in the Hardy inequality on $\mathbb{R}^n.$ We prove that there exists a threshold $ γ_{crit}(α)$ in $( - \infty, γ_H(α)) $ such that the existence of solutions of the above problem is guaranteed by the non-linear perturbation $(i.e., h(x) u^{q-1})$ whenever $ γ\le γ_{crit}(α),$ while for $γ_{crit}(α)<γ<γ_H(α)$, it is determined by a subtle combination of the geometry of the domain and the size of the nonlinearity of the perturbations.

math.AP↗

arXiv2017

A variational problem associated to a hyperbolic Caffarelli--Kohn--Nirenberg inequality

We prove a Caffarelli--Kohn--Nirenberg inequality in the hyperbolic space. For a semilinear elliptic equation involving the associated weighted Laplace--Beltrami operator, we establish variationally the existence of positive radial solutions in the subcritical regime. We also show a non-existence result in star-shaped domains when the exponent is supercritical.

math.AP↗

arXiv2017

Multiple Positive Solutions for Nonlocal Elliptic Problems Involving the Hardy Potential and Concave-Convex Nonlinearities

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ Δ})^{\fracα{2}}u- γ\frac{u}{|x|^α}= λf(x) |u|^{q - 2} u + g(x) {\frac{|u|^{p-2}u}{|x|^s}} \ \text{ in } {Ω,} \quad \text{ with Dirichlet boundary condition } u = 0 \ \text{ in } \mathbb{R}^n \setminus Ω,$$ where $Ω\subset \mathbb{R}^n$ is a smooth bounded domain in $\mathbb{R}^n$ containing $0$ in its interior, and $f,g \in C(\overlineΩ)$ with $f^+,g^+ \not\equiv 0$ which may change sign in $\overlineΩ.$ We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for $λ$ sufficiently small. The variational approach requires that $0 < α<2,$ $ 0 <s < α<n,$ $ 1<q<2<p \le 2_α^*(s):= \frac{2(n-s)}{n-α},$ and $ γ< γ_H(α) ,$ the latter being the best fractional Hardy constant on $\mathbb{R}^n.$

math.AP↗

arXiv2017

Mass and Asymptotics associated to Fractional Hardy-Schrödinger Operators in Critical Regimes

We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schrödinger operator $ L_{γ,α}: = ({-}{ Δ})^{\fracα{2}}- \fracγ{|x|^α}$ on domains of $\mathbb{R}^n$ containing the singularity $0$, where $0<α<2$ and $ 0 \le γ< γ_H(α)$, the latter being the best constant in the fractional Hardy inequality on $\mathbb{R}^n$. We tackle the existence of least-energy solutions for the borderline boundary value problem $(L_{γ,α}-λI)u= {\frac{u^{2^\star_α(s)-1}}{|x|^s}}$ on $Ω$, where $0\leq s <α<n$ and $ 2^\star_α(s)={\frac{2(n-s)}{n-α}}$ is the critical fractional Sobolev exponent. We show that if $γ$ is below a certain threshold $γ_{crit}$, then such solutions exist for all $0<λ<λ_1(L_{γ,α})$, the latter being the first eigenvalue of $L_{γ,α}$. On the other hand, for $γ_{crit}<γ<γ_H(α)$, we prove existence of such solutions only for those $λ$ in $(0, λ_1(L_{γ,α}))$ for which the domain $Ω$ has a positive {\it fractional Hardy-Schrödinger mass} $m_{γ, λ}(Ω)$. This latter notion is introduced by way of an invariant of the linear equation $(L_{γ,α}-λI)u=0$ on $Ω$.

math.AP↗

arXiv2015

Borderline variational problems involving fractional Laplacians and critical singularities

We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \begin{equation*} μ_{γ,s}(\R^n):= \inf\limits_{u \in H^{\fracα{2}} (\R^n)\setminus \{0\}} \frac{ \int_{\R^n} |({-}{ Δ})^{\fracα{4}}u|^2 dx - γ\int_{\R^n} \frac{|u|^2}{|x|^α}dx }{(\int_{\R^n} \frac{|u|^{2_α^*(s)}}{|x|^{s}}dx)^\frac{2}{2_α^*(s)}}, \end{equation*} where $0\leq s<α<2$, $n>α$, ${2_α^*(s)}:=\frac{2(n-s)}{n-α},$ and $γ\in \mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on $\R^n$, \begin{equation*} \left\{\begin{array}{lll} ({-}{ Δ})^{\fracα{2}}u- γ\frac{u}{|x|^α}&= |u|^{2_α^*-2} u + {\frac{|u|^{2_α^*(s)-2}u}{|x|^s}} & \text{in } {\R^n}\\ \hfill u&>0 & \text{in } \R^n, \end{array}\right. \end{equation*} where $2_α^*:=\frac{2 n}{n-α}$ is the critical $α$-fractional Sobolev exponent, and $γ< γ_H:=2^α\frac{Γ^2(\frac{n+α}{4})}{Γ^2(\frac{n-α}{4})}$, the latter being the best fractional Hardy constant on $\R^n$.

math.AP↗