相关论文: Fractional logarithmic Schr\"{o}dinger equations o…
In this paper, we study the following quasilinear {S}chr\"{o}dinger equation: $$\left\{ \begin{array}{l} - {\Delta u} - \frac{\kappa }{2}\Delta \left( {u}^{2}\right) u = h\left( u\right) \text{ in }{\mathbb{R}}^{N}, \\ u \in {H}^{1}\left(…
We investigate the following Kirchhoff-type biharmonic equation \begin{equation}\label{pr} \left\{ \begin{array}{ll} \Delta^2 u+ \left(a+b\int_{\mathbb{R}^N}|\nabla u|^2d x\right)(-\Delta u+V(x)u)=f(x,u),\quad x\in \mathbb{R}^N,\\ u\in…
In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, $$ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, $$ where $n \geq 1$,…
We consider the existence, non-existence and multiplicity of positive solutions to the following critical Hardy-H\'{e}non equation with logarithmic term \begin{equation*}\label{eq11}\left\{ \begin{array}{ll} -\Delta u…
We consider the fractional Schrodinger equation with a logarithmic nonlinearity, when the power of the Laplacian is between zero and one. We prove global existence results in three different functional spaces: the Sobolev space…
This paper deals with the following class of nonlocal Schr\"odinger equations $$ \displaystyle (-\Delta)^s u + u = |u|^{p-1}u \ \ \text{in} \ \mathbb{R}^N, \quad \text{for} \ s\in (0,1). $$ We prove existence and symmetry results for the…
We are mainly concerned with the nonlinear $p$-Laplace equation \begin{equation*} -\Delta_pu+\rho|u|^{p-2}u=\psi(x,u) \end{equation*} on a locally finite graph $G=(V,E)$, where $p$ belongs to $(1, +\infty)$. We obtain existence of positive…
We introduce a discrete-time fractional calculus of variations on the time scale $h\mathbb{Z}$, $h > 0$. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and…
In this work, we study the existence and multiplicity of solutions for the following problem \begin{equation}\label{probaa1} \left\{ \begin{aligned} -(\Delta)_{p}^{s} u + V(x)|u|^{p-2}u &= \lambda f(u),&x\in\Omega; u&=0,&x\in…
We consider the defocusing periodic fractional nonlinear Schr\"odinger equation $$ i \partial_t u +\left(-\Delta\right)^{\alpha}u=-\lvert u \rvert ^2 u, $$ where $\frac{1}{2}< \alpha < 1$ and the operator $(-\Delta)^\alpha$ is the…
This paper is divided in two parts. In the first part, we prove coercivity results and minimization of the Euler energy functional. In the second part, we focus on the existence and multiplicity of a positive solution of fractional…
In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb…
In this paper we analyze the semi-linear fractional Laplace equation $$(-\Delta)^s u = f(u) \quad\text{ in } \mathbb{R}^N_+,\quad u=0 \quad\text{ in } \mathbb{R}^N\setminus \mathbb{R}^N_+,$$ where $\mathbb{R}^N_+=\{x=(x',x_N)\in…
For $s \in (\frac{1}{2},1]$ we investigate well-posedness of the equation \[ \left ( i \partial_t + (-\Delta)^{s} \right ) u = \left (|D|^{1-2s} |u|^2 \right)\ |D|^{2s-1} u \] under small initial data…
This work is devoted to study the existence of infinitely many weak solutions to nonlocal equations involving a general integrodifferential operator of fractional type. These equations have a variational structure and we find a sequence of…
In this paper we are concerned with the construction of periodic solutions of the nonlocal problem $(-\Delta)^s u= f(u)$ in $\mathbb{R}$, where $(-\Delta)^s$ stands for the $s$-Laplacian, $s\in (0,1)$. We introduce a suitable framework…
We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i\partial_t u - \Lambda u = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3 \bar{u}^3,…
In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-\Delta)^{\frac{1}{2}}u + u &= Q(x)f(u)\;\;\mbox{in}\;\;\R…
We prove uniqueness of solutions to the Cauchy problem for the derivative nonlinear Schr\"odinger equation in $L^\infty_tH^{1/2}_x$. Our proof is based on the method of normal form reduction (NFR), which has been employed to obtain the…
We consider the computations of the action ground state for a rotating nonlinear Schr\"odinger equation. It reads as a minimization of the action functional under the Nehari constraint. In the focusing case, we identify an equivalent…