Related papers: De Giorgi type results for elliptic systems

We consider linearly stable elliptic fixed points for a symplectic vector field and prove generic results of super-exponential stability for nearby solutions. Morbidelli and Giorgilli have proved a theorem of stability over…

We prove that every bounded stable solution of \[ (-\Delta)^{1/2} u + f(u) =0 \qquad \mbox{in }\mathbb R^3\] is a 1D profile, i.e., $u(x)= \phi(e\cdot x)$ for some $e\in \mathbb S^2$, where $\phi:\mathbb R\to \mathbb R$ is a nondecreasing…

We investigate the H\'enon-Lane-Emden system defined by $- \Delta u=|x|^a |v|^{p-1}v$ and $- \Delta v=|x|^b |u|^{q-1}u$ in $\mathbb{R}^N \!\setminus\! \{0\}$. We begin by establishing a general Liouville-type theorem for the subcritical…

Let $(M^n,g)$ be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$\Delta u+au\log u=0,$$ where $a$ is a nonzero…

Flexibility and rigidity properties of steady (time-independent) solutions of the Euler, Boussinesq and Magnetohydrostatic equations are investigated. Specifically, certain Liouville-type theorems are established which show that suitable…

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…

In this paper we prove the validity of Gibbons' conjecture for a coupled competing Gross-Pitaevskii system. We also provide sharp a priori bounds, regularity results and additional Liouville-type theorems.

We study properties of the semilinear elliptic equation $\Delta u = 1/u$ on domains in $R^n$, with an eye toward nonnegative singular solutions as limits of positive smooth solutions. We prove the nonexistence of such solutions in low…

In this paper, we develop an elementary and unified treatment, in the spirit of Rivi\`ere and Struwe (Comm. Pure. Appl. Math. 2008), to explore regularity of weak solutions of higher order geometric elliptic systems in critical dimensions…

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…

In this paper we are concerned with the number of nonnegative solutions of the elliptic system $$ {array}{ll} -\Delta u = Q_u(u,v) + 1/2{2^*} H_u(u,v),& {in} \Omega,\vdois\ -\Delta v = Q_v(u,v) + 1/{2^*} H_v(u,v),& {in} \Omega,\vdois\…

We consider classical solutions to $-\Delta u = f(u)$ in half-spaces, under homogeneous Dirichlet boundary conditions. We prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary, provided…

We investigate Liouville-type results, existence, uniqueness and symmetry to the solution of nonlinear nonlocal elliptic equations of the form \[ Lu = |x|^{\gamma}\,H(u)\,G(\nabla u), \qquad x\in\R^n, \] where $L$ is a symmetric,…

We consider the elliptic quasilinear equation --$\Delta$ m u = u p |$\nabla$u| q in R N with q $\ge$ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N are constant. We also…

In this paper we study solutions, possibly unbounded and sign-changing, of the following problem: -\D_{\lambda} u=|x|_{\lambda}^a |u|^{p-1}u, in R^n,\;n\geq 1,\; p>1, and a \geq 0, where \D_{\lambda} is a strongly degenerate elliptic…

We examine the general weighted Lane-Emden system \begin{align*} -\Delta u = \rho(x)v^p,\quad -\Delta v= \rho(x)u^\theta, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N \end{align*} where $1<p\leq\theta$ and $\rho: \mathbb{R}^N\rightarrow…

In this paper, we study the Liouville-type equation \[\Delta ^2 u-\lambda_1\kappa\Delta u+\lambda_2\kappa^2(1-\mathrm e^{4u})=0\] on a closed Riemannian manifold \((M^4,g)\) with \(\operatorname{Ric}\geqslant 3\kappa g\) and \(\kappa>0\).…

In this paper, the Liouville-type theorems for the steady Navier-Stokes system are investigated. First, we prove that any bounded smooth helically symmetric solution in $\mathbb{R}^3$ must be a constant vector. Second, for steady…

In this short note we treat a 1+1-dimensional system of changing type. On different spatial domains the system is of hyperbolic and elliptic type, that is, formally, $\partial_t^2 u_n-\partial_x^2 u_n = \partial_t f$ and $u_n-\partial_x^2…

This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the…