Related papers: Multi-bump solutions for logarithmic Schr\"odinger…

This paper concerns with the existence of nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & = \gamma H_{e}(|u|-a)|u|^{q-2}u+|u|^{2^{*}-2}u\;\;\mbox{ in}\;\;\mathbb{R}^{N},\nonumber u…

In this paper, we consider the following nonlinear Schr\"odinger system in $R^3$: \begin{align*} -\Delta u_j +P_j(x) u=\mu_j u_j^3+\sum\limits_{i=1,i\neq j}^N\beta_{ij}u_i^2u_j, \end{align*} where $N\geq3$, $P_j$ are nonnegative radial…

We study, in the semiclassical limit, the singularly perturbed nonlinear Schr\"odinger equations $$ L^{\hbar}_{A,V} u = f(|u|^2)u \quad \mbox{in } R^N $$ where $N \geq 3$, $L^{\hbar}_{A,V}$ is the Schr\"odinger operator with a magnetic…

We study stable blow-up dynamics in the $L^2$-supercritical nonlinear Schr\"{o}dinger equation in various dimensions. We first investigate the profile equation and extend the result of X.-P. Wang [38] and Budd et al. [4] on the existence…

We construct time quasi-periodic solutions to the energy supercritical nonlinear Schr\"odinger equations on the torus in arbitrary dimensions. This introduces a new approach, which could have general applicability.

We study the influence of the nonlinearity in the Schrodinger equation on the motion of quantum particles in a harmonic trap. In order to obtain exact analytic solutions, we have chosen the logarithmic nonlinearity. The unexpected result of…

In this paper, we prove the existence of positive solutions $(\lambda_1,\lambda_2, u,v)\in \R^2\times H^1(\R^N, \R^2)$ to the following coupled Schr\"odinger system $$\begin{cases} -\Delta u + \lambda_1 u= \mu_1|u|^{p-2}u+\beta v \quad…

We investigate the nonlinear Schr\"{o}dinger equation $iu_{t}+\Delta u+|u|^{p-1}u=0$ with $1+\frac{4}{N}<p<1+\frac{4}{N-2}$ (when $N=1, 2$, $1+\frac{4}{N}<p<\infty$) in energy space $H^1$ and study the divergent property of…

We study the Calder\'on problem for a logarithmic Schr\"odinger type operator of the form $L_{\Delta} +q$, where $L_{\Delta}$ denotes the logarithmic Laplacian, which arises as formal derivative $\frac{d}{ds} \big|_{s=0}(-\Delta)^s$ of the…

In this paper, we study critical semilinear nonlocal elliptic equations involving the logarithmic Schr\"odinger operator and its fractional pseudo-relativistic counterpart, both arising in quantum models with nonlocal and relativistic…

In this paper, we study the local well-posedness of the cubic Schr\"odinger equation: \[ (i \partial_t - \mathscr{L}) u = \pm |u|^2 u \quad \text{ on } I \times \mathbb{R}^d, \] with randomized initial data, and $\mathscr{L}$ being an…

In this paper, we apply the Ljusternik-Schnirelman theory with local Palais-Smale condition to study a class of N-body problems with strong force potentials and fixed energies. Under suitable conditions on the potential $V$, we prove the…

We consider the system of coupled elliptic equations \[ \begin{cases} -\Delta u - \lambda_1 u = \mu_1 u^3+ \beta u v^2 \\ -\Delta v- \lambda_2 v = \mu_2 v^3 +\beta u^2 v \end{cases} \text{in $\mathbb{R}^3$}, \] and study the existence of…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

We study the nonlinear Schr\"odinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially…

It is shown that using the similarity transformations, a set of three-dimensional p-q nonlinear Schrodinger (NLS) equations with inhomogeneous coefficients can be reduced to one-dimensional stationary NLS equation with constant or varying…

We revisit the following nonlinear critical elliptic equation \begin{equation*} -\Delta u+Q(y)u=u^{\frac{N+2}{N-2}},\;\;\; u>0\;\;\;\hbox{ in } \mathbb{R}^N, \end{equation*} where $N\geq 5.$ There seems to be no results about the…

We study the Schr\"odinger equations $-\Delta u + V(x)u = f(x,u)$ in $\mathbb{R}^N$ and $-\Delta u - \lambda u = f(x,u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$. We assume that $f$ is superlinear but of subcritical growth and…

In this paper, we study the fractional logarithmic Schr\"{o}dinger equation $$ (-\Delta)^{s} u+h(x) u=u \log u^{2} $$ on lattice graphs $\mathbb{Z}^d$, where $s\in (0,1)$. If $h(x)$ is a bounded periodic potential, we prove the existence of…

In this paper we consider the following class of elliptic problems $$- \Delta_A u + u = a_{\lambda}(x) |u|^{q-2}u+b_{\mu}(x) |u|^{p-2}u ,\,\, x\in \mathbb{R}^N$$ where $1<q<2<p<2^*-1= \frac{N+2}{N-2}$, $a_{\lambda}(x)$ is a sign-changing…