Related papers: Concentration Behavior of Nonlinear Hartree-type E…

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…

We study the global behavior of solutions to the nonlinear generalized Hartree equation, where the nonlinearity is of the non-local type and is expressed as a convolution, $$ i u_t + \Delta u + (|x|^{-(N-\gamma)} \ast |u|^p)|u|^{p-2}u=0,…

We consider the following time-independent nonlinear $L^2$-critical Schr\"{o}dinger equation \[ -\Delta u(x)+V(x)u(x)-a|x|^{-b}|u|^{1+\frac{4-2b}{N}}=\mu u(x)\,\ \hbox{in}\,\ \mathbb{R}^N, \] where $\mu\in\mathbb{R}$, $a>0$, $N\geq 1$,…

We study the following nonlinear Schr\"{o}dinger equation $$ iu_t=-\Delta u+V(x)u-a|u|^qu \quad (t,x)\in \mathbb{R}^1\times \mathbb{R}^2, $$ where $a>0, \ q\in(0,2)$ and $V(x)$ is some type of trapping potentials. For any fixed $a>a^*:=…

In this paper we are interested to prove the existence and concentration of ground state solution for the following class of problems $$ -\Delta u+V(x)u=A(\epsilon x)f(u), \quad x \in \R^{N}, \eqno{(P)_{\epsilon}} $$ where $N \geq 2$,…

With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[\sqrt{-\Delta+m^2}\, u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $V$ is a bounded…

In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like $$ \left\{\begin{array}{l} -\Delta u+V(x)u=A(\epsilon x)f(u) \quad \mbox{in} \quad \R^{N}, \\ u\in…

In this paper, we study the following coupled nonlinear logarithmic Hartree system \begin{align*} \left\{ \displaystyle \begin{array}{ll} \displaystyle -\Delta u+ \lambda_1 u =\mu_1\left( -\frac{1}{2\pi}\ln(|x|) \ast u^2 \right)u+\beta…

In this paper, we consider the dynamics of the solution to the mass critical focusing Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$. The main difficulties are the equation is \emph{not}…

We study the asymptotic profile, as $\hbar\rightarrow 0$, of positive solutions to $$-\hbar^2\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta u^2=K(x)|u|^{p-2}u,\ \ x\in \mathbb{R}^N $$ where $\gamma\geq 0$ is a parameter with relevant physical…

In this paper we investigate the isolated singularities of the Hartree type equation \begin{equation*} -\Delta u (x)= \left(\frac{1}{|x|^\alpha}*e^u\right)e^{u(x)}\quad \text{in } B_{1}\setminus\{0\} , \end{equation*} where $\alpha>0$,…

We consider here solutions of the nonlinear fractional Schr\"odinger equation $$\epsilon^{2s}(-\Delta)^s u+V(x)u=u^p.$$ We show that concentration points must be critical points for $V$. We also prove that, if the potential $V$ is coercive…

In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation $$ \left\{\begin{array}{ll} -(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u=\mu…

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…

We study the asymptotic behavior of solutions to the nonlocal nonlinear equation $(-\Delta_p)^s u=|u|^{q-2}u$ in a bounded domain $\Omega\subset{\mathbb R}^N$ as $q$ approaches the critical Sobolev exponent $p^*=Np/(N-ps)$. We prove that…

In this paper, we study the asymptotic behavior of energy and action ground states to the following pseudo-relativistic Hartree equation \[ \left(\sqrt{-c^2\Delta +m^2c^4}-mc^2\right)u + \lambda u = \left(|x|^{-1}*|u|^2\right)u \] as the…

In this work, we consider the focusing generalized inhomogeneous Hartree equation with potential \[ i u_t + \Delta u - V(x)u + \left(I_{\gamma} * |x|^{-b}|u|^{p}\right)|x|^{-b}|u|^{p-2}u = 0, \] where $0<\gamma<3$ and…

We study the concentration phenomenon for solutions of the fractional nonlinear Schr\"{o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation…

This paper is concerned with the ground states of a coupled pseudo-relativistic Hartree system in $\mathbb{R} ^{3} $ with trapping potentials, where the intraspecies and the interspecies interaction are both attractive. By investigating an…

We are interested in the general Choquard equation \begin{multline*} \sqrt{\strut -\Delta + m^2} \ u - mu + V(x)u - \frac{\mu}{|x|} u = \left( \int_{\mathbb{R}^N} \frac{F(y,u(y))}{|x-y|^{N-\alpha}} \, dy \right) f(x,u) - K (x) |u|^{q-2}u…