相关论文: Limit Behavior of Mass Critical Hartree Minimizati…

In this paper, we investigate the constrained minimization problem \begin{equation}\label{eq:0.1} e(a):=\inf_{\{u\in \mathcal{H},\|u\|_2^2=1\}}E_a(u), \end{equation} where the energy functional \begin{equation} \label{eq:0.2}…

In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: $$ E(u)=\frac{1}{2}\ds\int_{\R^N}|\nabla…

Let $a>0,b>0$ and $V(x)\geq0$ be a coercive function in $\mathbb R^2$. We study the following constrained minimization problem on a suitable weighted Sobolev space $\mathcal{H}$: \begin{equation*}…

We are concerned with the following constrained minimization problem: $$e(a_{1},a_{2},\beta) := \inf\left\{E_{a_{1},a_{2},\beta}(u_{1},u_{2}): \|u_{1}\|_{L^{2}(\mathbb{R}^{3})} = \|u_{2}\|_{L^{2}(\mathbb{R}^{3})} = 1\right\},$$ where…

This paper is devoted to the $L^2$-constraint variational problem \begin{equation*} We study $L^2$-normalized solutions of the following inhomogeneous fractional Schr\"{o}dinger equation \begin{equation*} (-\Delta)^{s}…

Inspired by Lin-Pan-Wang (Comm. Pure Appl. Math., 65(6): 833-888, 2012), we continue to study the corresponding time-independent case of the Keller-Rubinstein-Sternberg problem. To be precise, we explore the asymptotic behavior of…

We study the existence and blow-up behavior of minimizers for $E(b)=\inf\Big\{\mathcal{E}_b(u) \,|\, u\in H^1(R^2), \|u\|_{L^2}=1\Big\},$ here $\mathcal{E}_b(u)$ is the Kirchhoff energy functional defined by $\mathcal{E}_b(u)= \int_{R^2}…

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…

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, using a discrete Schwarz rearrangement on lattice graphs developed in \cite{DSR}, we study the existence of global minimizers for the following functional $I:H^1\left(\mathbb{Z}^N\right)\to \R$, $$I(u)=\frac{1}{2}…

This paper is concerned with the following focusing biharmonic HLS upper critical Hartree equation with a local perturbation $$ \begin{cases} {\Delta}^2u-\lambda u-\mu|u|^{p-2}u-(I_\alpha*|u|^{4^*_\alpha})|u|^{4^*_\alpha-2}u=0\ \ \mbox{in}\…

We study minimizers of the pseudo-relativistic Hartree functional $$\mathcal{E}_{a}(u):=\|(-\Delta+m^{2})^{1/4}u\|_{L^{2}}^{2}-\frac{a}{2}\int_{\mathbb{R}^{3}}(\left|\cdot\right|^{-1}\star |u|^{2})(x)|u(x)|^{2}{\rm…

We consider a small random perturbation of the energy functional $$ [u]^2_{H^s(\Lambda, R^d)} + \int_\Lambda W(u(x)) dx $$ for $s \in (0,1),$ where the non-local part $ [u]^2_{H^s(\Lambda,R^d)}$ denotes the total contribution from $\Lambda…

In any dimension $N \geq 1$, for given mass $m > 0$ and for the $C^1$ energy functional \begin{equation*} I(u):=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx-\int_{\mathbb{R}^N}F(u)dx, \end{equation*} we revisit the classical problem of…

We study the non-linear minimization problem on $H^1_0(\Omega)\subset L^q$ with $q=\frac{2n}{n-2}$, $\alpha>0$ and $n\geq4$~: \[\inf_{\substack{u\in H^1_0(\Omega) \|u\|_{L^q}=1}}\int_\Omega a(x,u)|\nabla u|^2 - \lambda \int_{\Omega}…

In this paper, we study the existence of minimizers for a class of constrained minimization problems derived from the Schr\"{o}dinger-Poisson equations: $$-\Delta u+V(x)u+(|x|^{-1}*u^2)u-|u|^\frac{4}{3}u=\lambda u,~~x\in\R^3$$ on the…

In this paper, we study the existence and the concentration behavior of minimizers for $i_V(c)=\inf\limits_{u\in S_c}I_V(u)$, here $S_c=\{u\in H^1(\R^N)|~\int_{\R^N}V(x)|u|^2<+\infty,~|u|_2=c>0\}$ and…

We are concerned with $L^2$-constraint minimizers for the Kirchhoff functional $$ E_b(u)=\int_{\Omega}|\nabla u|^2\mathrm{d}x+\frac{b}{2}\left(\int_\Omega|\nabla u|^2\mathrm{d}x\right)^2+\int_\Omega…

We propose a simple minimization method to show the existence of least energy solutions to the normalized problem \begin{cases} -\Delta u + \lambda u = g(u) \quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, \\ u \in H^1(\mathbb{R}^N), \\…

We study the following nonlinear Hartree-type equation \begin{equation*} -\Delta u+V(x)u-a(\frac{1}{|x|^\gamma}\ast |u|^2)u=\lambda u,~\text{in}~\mathbb{R}^N, \end{equation*} where $a>0$, $N\geq3$, $\gamma\in(0,2)$ and $V(x)$ is an external…