Related papers: Standing waves for nonlinear Hartree type equation…

This paper is motivated by a gauged Schr\"{o}dinger equation in dimension 2. We are concerned with radial stationary states under the presence of a vortex at the origin. Those states solve a nonlinear nonlocal PDE with a variational…

In this note we study analytically and numerically the existence and stability of standing waves for one dimensional nonlinear Schr\"odinger equations whose nonlinearities are the sum of three powers. Special attention is paid to the curves…

Consider the following semilinear problem with a point interaction in $\mathbb{R}^N$: \[- \Delta_\alpha u + \omega u = u |u|^{p - 2},\] where $N \in \{2, 3\}$; $\omega > 0$; $- \Delta_\alpha$ denotes the Hamiltonian of point interaction…

We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_\alpha\ast u^p)u^q$ in $\Omega$, where $\Omega\subset \mathbb{R}^N,…

We study the system $$ \left\{ -\Delta u+u+K(x) \phi |u|^{q-2}u&=(I_\alpha*|u|^p)|u|^{p-2}u &&\mbox{ in }{\mathbb R}^N, -\Delta \phi&=K(x)|u|^q&&\mbox{ in }{\mathbb R}^N, \right. $$ where $N\geq 3$, $\alpha\in (0,N)$, $p,q>1$ and $K\geq 0$.…

In this work, we study the existence of various classes of standing waves for a nonlinear Schr\"odinger system with quadratic interaction, along with a harmonic or partially harmonic potential. We establish the existence of ground-state…

We study the existence and stability of standing waves solutions of a three-coupled nonlinear Schr\"{o}dinger system related to the Raman amplification in a plasma. By means of the concentration-compacteness method, we provide a…

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation in $H^1(\mathbb{R}^3)$, \begin{equation} i\partial_t u + \Delta u + |x|^{-b}|u|^{2}u=0,{equation} where $0 < b <\tfrac{1}{2}$. Previous works have established a…

We study a class of two-dimensional non-linear Schr\"odinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces…

We show that solutions $u(x,t)$ of the non-stationnary incompressible Navier--Stokes system in $\R^d$ ($d\geq2$) starting from mild decaying data $a$ behave as $|x|\to\infty$ as a potential field: u(x,t) = e^{t\Delta}a(x) +…

In this paper, we study the following Kirchhoff type problem:% $$ \left\{\aligned&-\bigg(\alpha\int_{\bbr^3}|\nabla u|^2dx+1\bigg)\Delta u+(\lambda a(x)+a_0)u=|u|^{p-2}u&\text{ in }\bbr^3,\\%…

In this paper we prove the existence of vortices, namely standing waves with non null angular momentum, for the nonlinear Klein-Gordon equation in dimension $N\geq 3$. We show with variational methods that the existence of these kind of…

In this paper, we consider a wave equation with strong damping and logarithmic nonlinearity. This paper aims to study the local and global existence, uniqueness and the uniform energy decay rate of a weak solution under some sufficient…

By employing the N-barrier method developed in the paper, we establish a new N-barrier maximum principle for diffusive Lotka-Volterra systems of two competing species. As an application of this maximum principle, we show under certain…

We consider the inhomogeneous nonlinear Schr\"odinger equation with inverse-square potential in $\mathbb{R}^N$ $$ i u_t + \mathcal{L}_a u+\lambda |x|^{-b}|u|^\alpha u = 0,\;\;\mathcal{L}_a=\Delta -\frac{a}{|x|^2}, $$ where $\lambda=\pm1$,…

We study the instability of standing waves for nonlinear Schr\"{o}dinger equations. Under a general assumption on nonlinearity, we prove that linear instability implies orbital instability in any dimension. For that purpose, we establish a…

We consider the Choquard system with both linear and nonlinear couplings $-\Delta u + \mu_1 u =\lambda_1 ( I_\alpha * |u|^{r_1} ) |u|^{r_1-2} u + \beta p( I_\alpha * |v|^q)|u|^{p-2} u + \kappa v,$ $-\Delta v + \mu_2 v =\lambda_2 ( I_\alpha…

We consider a system of nonlinear Klein-Gordon equations with quadratic interaction in two and three space dimensions. The strong instability of standing wave solutions is studied for the system without assuming the mass resonance…

Rotating waves are a fascinating feature of a wide array of complex systems, particularly those arising in the study of many chemical and biological processes. With many rigorous mathematical investigations of rotating waves relying on the…

This paper is the second part of a work devoted to the study of elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\begin{cases} -\Delta u-\lambda \frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=\kappa\alpha…