相关论文: Qualitative properties of single blow-up solutions…
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…
In this paper, we are concerned with the critical elliptic equation \begin{equation}\label{kx} \left\lbrace\begin{aligned} &-\Delta u=u^{p}+\epsilon \kappa(x)u^{q}\quad\hspace{2mm} \mbox{in}~~\Omega, \\&u>0\quad…
In this paper, we study that the nearly critical nonlocal problem \begin{equation*} \left\lbrace \begin{aligned} &-\Delta u=(|x|^{-{(n-2)}}\ast u^{p-\epsilon})u^{p-1-\epsilon} \quad \mbox{in}\quad \Omega, &u>0\quad…
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…
In this paper, we are concerned with qualitative properties of multi-peak solutions of the following nonlinear Schr\"{o}dinger equations \begin{equation*} -\Delta u+V(x)u= u^{p-\varepsilon},\,\,\,u>0,\,\,\,\text{in}\,\,\,\mathbb{R}^N,…
We investigate the qualitative properties of a critical Hartree equation defined on punctured domains. Our study has two main objectives: analyzing the asymptotic behavior near isolated singularities and establishing radial symmetry of…
We study the generalized Hartree equation, which is a nonlinear Schr\"odinger-type equation with a nonlocal potential $iu_t + \Delta u + (|x|^{-b} \ast |u|^p)|u|^{p-2}u=0, x \in \mathbb{R}^N$.We establish the local well-posedness at the…
We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, a Schr\"odinger-type equation with a nonlocal, convolution-type nonlinearity: $iu_t+\Delta u +\left(|x|^{-(d-2)} \ast |u|^{p} \right) |u|^{p-2}u = 0,…
In this paper, we study the qualitative behavior at a vortex blow-up point for Chern-Simon-Higgs equation. Roughly speaking, we will establish an energy identity at a each such point, i.e. the local mass is the sum of the bubbles. Moreover,…
We prove some qualitative properties for singular solutions to a class of strongly coupled system involving a Gross--Pitaevskii-type nonlinearity. Our main theorems are vectorial fourth order counterparts of the classical results of [J.…
We study the asymptotic behavior of a sequence of positive solutions $(u_{\epsilon})_{\epsilon >0}$ as $\epsilon \to 0$ to the family of equations \begin{equation*} \left\{\begin{array}{ll} \Delta u_{\epsilon}+a(x)u_{\epsilon}=…
We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-\Delta u= \lambda V(x) e^u-4\pi N \delta_0\;\mbox{ in } B_1,\quad u=0 \;\mbox{ on }\partial B_1,$$ where $B_1$ is the unit ball in…
In this paper, we study the following Lane-Emden system with nearly critical non-power nonlinearity \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{lll} -\Delta u =\frac{|v|^{p-1}v}{[\ln(e+|v|)]^\epsilon}\ \ &{\rm in}\ \Omega,…
We consider the slightly subcritical elliptic problem with Hardy term $$ \left\{ \begin{aligned} -\Delta u-\mu\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\epsilon}u &&\quad \text{in } \Omega\subset\mathbb{R}^N, \\\ u &= 0&&\quad \text{on } \partial…
We consider the following nonlinear Schr\"{o}dinger equation with an inverse potential: \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2\sigma}}\log|x|u=0 \] in $\mathbb{R}^N$. From the classical argument, the…
The objective of this paper is to obtain qualitative characteristics of multi-bubble solutions to the Lane-Emden-Fowler equations with slightly subcritical exponents given any dimension $n \ge 3$. By examining the linearized problem at each…
We consider the following nonlinear Schr\"{o}dinger equation with a Hartree nonlinearity: \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u\pm\left(\frac{1}{|x|^{2\sigma}}\star|u|^2\right)u=0 \] in $\mathbb{R}^N$. We are…
We are concerned with the qualitative analysis of positive singular solutions with blow-up boundary for a class of logistic-type equations with slow diffusion and variable potential. We establish the exact blow-up rate of solutions near the…
We characterize the dynamics of the finite time blow up solutions with minimal mass for the focusing mass critical Hartree equation with $H^1(\mathbb{R}^4)$ data and $L^2(\mathbb{R}^4)$ data, where we make use of the refined…
We discuss the H\'{e}non parabolic equation $\partial_t u = \Delta u + |x|^\sigma u^p$ in a finite ball in $\mathbb{R}^N$ under the Dirichlet boundary condition, where $N\ge1$, $p>1$, and $\sigma>0$. We assume that the exponent $p$ is…