Related papers: A Liouville-type theorem for Schr\"odinger equatio…

In this paper, we study the existence, uniqueness, nondegeneracy and some qualitative properties of positive solutions for the logarithmic Schr\"odinger equations: \[ -\Delta u+ V(|x|) u=u\log u^2, u\in H^1(\mathbb R^N). \] Here $N\geq 2$…

In this paper, we consider global solutions of the following nonlinear Schr\"odinger equation $iu_t+\Delta u+\lambda|u|^\alpha u = 0,$ in $\R^N,$ with $\lambda\in\R,$ $\alpha\in(0,\frac{4}{N-2})$ $(\alpha\in(0,\infty)$ if $N=1)$ and…

In this paper, we consider the following Schr\"odinger-Poisson system \begin{equation*} \begin{cases} - \Delta u+\lambda V(x)u+ \mu\phi u=|u|^{p-2}u &\text{in $\mathbb{R}^3$},\cr -\Delta \phi=u^{2} &\text{in $\mathbb{R}^3$}, \end{cases}…

In this paper, we deal with the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t = \Delta u + 2uh'(|u|^2)\Delta h(|u|^2) + V(x)u,\ x\in \mathbb{R}^N,\ t>0\\ u(x,0) = u_0(x), \quad x \in \mathbb{R}^N.…

In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum…

Let $N\geq 2$ and $1 < p < (N+2)/(N-2)_{+}$. Consider the Lane-Emden equation $\Delta u + u^p = 0$ in $\mathbb{R}^N$ and recall the classical Liouville type theorem: if $u$ is a non-negative classical solution of the Lane-Emden equation,…

This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$ \partial_t u (x,t) = \sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \qquad (x,t) \in \mathbb{R}^N \times\, ]- \infty…

We study the second order nonlinear differential equation \begin{equation*} u"+ \sum_{i=1}^{m} \alpha_{i} a_{i}(x)g_{i}(u) - \sum_{j=0}^{m+1} \beta_{j} b_{j}(x)k_{j}(u) = 0, \end{equation*} where $\alpha_{i},\beta_{j}>0$, $a_{i}(x),…

In this paper we prove Liouville type theorem for the stationary Navier-Stokes equations in $\Bbb R^3$ under the assumptions on the relative decays of velocity, pressure and the head pressure. More precisely, we show that any smooth…

We prove new one-dimensional symmetry results for non-negative solutions, possibly unbounded, to the semilinear equation $ -\Delta u= f(u)$ in the upper half-space $\mathbb{R}^{N}_{+}$. Some Liouville-type theorems are also proven in the…

We consider the asymptotics of the one-dimensional cubic nonlinear Schr\"odinger equation with an external potential $V$ that does not admit bound states. Assuming that $\jBra{x}^{2+}V(x) \in L^1$ and that $u$ is orthogonal to any…

We consider the following Schr\"{o}dinger equation $$ - \hslash ^2 \Delta u + V(x)u = \Gamma(x) f(u) \quad \mathrm{in} \ \mathbb{R}^N, $$ where $u \in H^1 (\mathbb{R}^N)$, $u > 0$, $\hslash > 0$ and $f$ is superlinear and subcritical…

We obtain a new Liouville comparison principle for entire weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$ u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*) $$…

In this paper we prove a Liouville type theorem for generalized stationary Navier-Stokes systems in $\Bbb R^3$, which model non-Newtonian fluids, where the Laplacian term $\Delta u$ is replaced by the corresponding non linear operator…

We prove a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators. We then apply the result in order to obtain gradient boundary blow up rates for…

We study the following nonlinear Schr\"odinger equation $$-\Delta u + V(x) u = g(x,u),$$ where V and g are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of $-\Delta+V$. The superlinear and subcritical…

We prove existence of positive ground state solutions to the pseudo-relativistic Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} \sqrt{-\Delta +m^2} u +Vu = \left( W * |u|^{\theta} \right)|u|^{\theta -2} u \quad\text{in…

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator $P_1$, such that a nonzero subsolution of a symmetric nonnegative operator $P_0$ is a ground state. Particularly, if…

We study $\mathbf L^\infty$ entropy solutions to $2\times 2$ systems of conservation laws. We show that, if a uniformly convex entropy exists, these solutions satisfy a pair of kinetic equations (nonlocal in velocity), which are then shown…

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega$ in $R^N$ with Dirichlet boundary conditions. The operator $L$ is a uniformly elliptic operator of order $2m$. We assume that for…