Related papers: Nondegeneracy of solutions for a critical Hartree …

The aim of this paper is to classify the positive solutions of the nonlocal critical equation: $$ -\Delta u=\left(I_{\mu}\ast u^{2^{\ast}_{\mu}}\right)u^{{2}^{\ast}_{\mu}-1},~~x\in\mathbb{R}^{N}, $$ where $0<\mu<N$, if $N=3\ \hbox{or} \ 4$…

In this paper, we are concerned with the critical Hartree equation \begin{equation*} \begin{cases} -\Delta u=\left(\displaystyle{\displaystyle{\int_{\Omega}}}\frac{u^{2^{*}_{\mu}}(y)}{|x-y|^{\mu}}dy\right)u^{2^{*}_{\mu}-1}+\varepsilon…

We study the uniqueness and nondegeneracy of positive bubble solutions for the generalized energy-critical Hartree equation on the Heisenberg group $\mathbb{H}^{n}$, \begin{equation}\label{0.1}…

In this paper, we are interested in some problems related to the following biharmonic hartree equation \begin{equation*} \Delta^{2} u=(|x|^{-\alpha}\ast |u|^{p})u^{p-1},\sp \text{in}\quad\R^N. \end{equation*} where…

In this paper, we prove uniqueness and nondegeneracy of positive solutions to the following Kirchhoff equations with critical growth \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u=u^{5}, & u>0 & \text{in…

We study the coupled Hartree system $$ \left\{\begin{array}{ll} -\Delta u+ V_1(x)u =\alpha_1\big(|x|^{-4}\ast u^{2}\big)u+\beta \big(|x|^{-4}\ast v^{2}\big)u &\mbox{in}\ \mathbb{R}^N,\\[1mm] -\Delta v+ V_2(x)v =\alpha_2\big(|x|^{-4}\ast…

We prove that any positive solution for the critical nonlinear Hartree equation $$-\Laplacian\fct{u}{x} -\int_{\R^6} \frac{\abs{\fct{u}{y}}^2 }{ \abs{x-y}^4 }\odif{y} \,\fct{u}{x}=0,\qtq{} x\in\R^6.$$ is nondegenerate. Firstly, in terms of…

We study nondegeneracy of ground states of the Hartree equation $$ -\Delta u+u=(I_{2}\ast u^2)u\quad\mbox{ in }\mathbb R^n $$ where $n=3,4,5$ and $I_2$ is the Newton potential. As an application of the nondegeneracy result, we use a…

This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent \begin{equation*} \begin{cases} -\Delta u+(V_1(x)+\lambda_1)u=\mu_1(|x|^{-4}*u^{2})u+\beta (|x|^{-4}*v^{2})u, \ \ &x\in R^N, -\Delta…

In this article, we establish the existence of solutions to the following critical Hartree equation \begin{align*} \begin{cases} -\Delta u=\left(\int_{\Omega_\varepsilon}\frac{u^{2_{\mu}^*}}{|x-y|^{\mu}}dy\right)u^{2_{\mu}^*-1}, &\text{ in…

This paper deals with the following equation $$-\Delta u =K(|x'|, x'')\Big(|x|^{-\alpha}\ast (K(|x'|, x'')|u|^{2^{\ast}_{\alpha}})\Big) |u|^{2^{\ast}_{\alpha}-2}u\quad\mbox{in}\ \mathbb{R}^N,$$ where $N\geq5$, $\alpha>5-\frac{6}{N-2}$,…

In this paper, we show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH) \begin{equation*} -{\Delta u}\sts{x} -{\bm\alpha}\sts{N,\lambda} \int_{\R^N} { \frac{…

In this paper we are interested in the following critical Hartree equation \begin{equation*} \begin{cases} -\Delta u =\displaystyle{\Big(\int_{\Omega}\frac{u^{2_{\mu}^\ast} (\xi)}{|x-\xi|^{\mu}}d\xi\Big)u^{2_{\mu}^\ast-1}}+\varepsilon u…

This paper is concerned with the quantitative stability of critical points of the Hardy-Littlewood-Sobolev inequality. Namely, we give quantitative estimates for the Choquard equation: $$-\Delta u=(I_{\mu}\ast|u|^{2_\mu^*}) u^{2_\mu^*-1}\ \…

This paper investigates the critical behavior of global solutions to a parabolic equation with a Hartree-type nonlinearity of the form $$\left\{\begin{array}{ll} u_{t}+(-\Delta)^{\frac{\beta}{2}} u= (\mathcal{K}\ast |u|^{p})|u|^{q},&\qquad…

In this paper we study a class of critical Choquard equations with a symmetric potential, i.e. we consider the equation $$-\Delta u +V(|x|) u =\left(|x|^{-\mu}* |u|^{2^\star_\mu}\right)|u|^{2^\star_\mu-2}u,\quad\mbox{in}\quad\mathbb R^N$$…

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy-Sobolev fractional equation with critical growth \begin{equation*}\label{0.1} \left\{% \begin{array}{ll} (-\Delta)^{s} u-\ds\frac{\mu…

The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-\Delta u =\frac{1}{|x|^{\alpha}}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{\alpha,…

We prove that positive solutions $u\in H^s(\mathbb{R}^N)$ to the equation $(-\Delta )^s u+ u=u^p$ in $\mathbb{R}^N$ are nonradially nondegenerate, for all $s\in (0,1)$, $N\geq 1$ and $p>1$ strictly smaller than the critical Sobolev…

This paper presents some qualitative properties of positive solutions to the strongly coupled system \[ \begin{cases} \displaystyle - \Delta u + \tau u = \frac{2 p}{p + q} \left( I_\alpha \ast |v|^q \right) |u|^{p - 2} u &\text{in} ~…