相关论文: Solutions of Liouville equations with non-trivial …
We study nonnegative solutions to the following Hardy-H\'enon type equations involving higher order fractional Laplacians $$ (-\Delta)^\sigma u = |x|^{-\alpha}u^{p} ~~~~~~ \mbox{in} ~ \mathbb{R}^n \backslash \{0\} $$ with a possible…
In the paper, a Liouville theorem for mild bounded ancient solutions to the 2D Navier-Stokes equations in half space has been proven.
We consider variational problems with regular H{\"o}lderian weight or with weight and boundary singularity and, Dirichlet condition. We prove the boundedness of the volume of the solutions to these equations on the annulus.
Small perturbation of the Liouville equation under smooth initial data is considered. Asymptotic solution which is available for a long time interval is constructed by the two scale method.
We investigate the initial-value problem of the non-linear Liouville hierarchy. For the general form of the interaction potential we construct an explicit solution in terms of an expansion over particle clusters whose evolution is described…
In this paper, we study quadratic growth solutions $u$ of fully nonlinear elliptic equations of the form $F(D^2u)=f$ in $\mathbb{R}^n$, where $f$ is periodic and $F$ may be not uniformly elliptic. The existence of solutions and Liouville…
We consider the problem of solvability of linear differential equations over a differential field~$K$. We introduce a class of special differential field extensions, which widely generalizes the classical class of extensions of differential…
We study the existence of solutions to the problem $$ (-\Delta)^{\frac{n}{2}}u = Qe^{nu}\quad\text{in }\mathbb{R}^n, \quad V := \int_{\mathbb{R}^n}e^{nu}dx < \infty,$$ where $Q=(n-1)!$ or $Q=-(n-1)!$. Extending the works of Wei-Ye and…
We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems: (i) a representation formula for classical solutions, (ii) a quantitative decay rate at which the…
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…
We provide a simple method for obtaining new Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. To illustrate the method we prove Liouville theorems (guaranteeing nonexistence of positive…
We study the Klein-Gordon equation in one spatial and one temporal dimension. Physically, this equation describes the wave function of a relativistic spinless boson with positive rest mass. Mathematically, this is the most elementary…
We revisit rescaling methods for nonlinear elliptic and parabolic problems and show that, by suitable modifications, they may be used for nonlinearities that are not scale invariant even asymptotically and whose behavior can be quite far…
We examine the following fourth order H\'enon equation \label{pipe} \Delta^2 u = |x|^\alpha u^p \qquad \text{in}\ \IR^N, where $ 0 < \alpha$. Define the Hardy-Sobolev exponent $ p_4(\alpha):= \frac{N+4 + 2 \alpha}{N-4}$. We show that in…
We consider here the simplified Ericksen-Leslie system on the whole three-dimensional space. This system deals with the incompressible Navier-Stokes equations strongly coupled with a harmonic map flow which models the dynamical behavior for…
This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at $0$ and $\infty$…
In this paper we construct a family of exact strong solutions to the two-dimensional incompressible liquid crystal equations with finite energy. The initial velocity is chosen to be rotationally symmetric and the image of the initial…
We prove some general results on the existence and uniqueness of solutions to the Liouville equation. Then, we discuss the sharpness and possible generalizations. Finally, we give several applications, arising in both mathematics and…
We give applications of known and new Liouville type theorems to universal singularity and decay estimates for non scale invariant elliptic problems, including Lane-Emden and Schr\"odinger type systems. This applies to various classes of…
In this paper, we provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the…