相关论文: Blowing-up solutions for a nonlocal Liouville type…

In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-\Delta)^\frac12 u= \kappa e^u-1~\mbox{in $S^1$,} \end{equation} where $(-\Delta)^\frac{1}{2}$ stands for the…

We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-\Delta u= \lambda V(x) e^u-4\pi N \delta_0\;\mbox{ in } B_1,\quad u=0 \;\mbox{ on }\partial B_1,$$ where $B_1$ is the unit ball in…

We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-\Delta u= \la a(x) e^u-4\pi N \delta_0\;\hbox{ in } \Omega,\quad u=0 \;\hbox{ on }\partial \Omega,$$ where $\Omega$ is a smooth and…

We consider a non-local Liouville equation corresponding to the prescription of the geodesic curvature on the circle. We build a family of solutions which blow up at a critical point of the harmonic extension of the prescribed curvature…

In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension $1$. More precisely, given a sequence $u_k :\mathbb{R} \to \mathbb{R}$ of solutions to \begin{equation} (-\Delta)^\frac{1}{2} u_k…

Let $m\ge 2$ be an integer. For any open domain $\Omega\subset\mathbb{R}^{2m}$, non-positive function $\varphi\in C^\infty(\Omega)$ such that $\Delta^m \varphi\equiv 0$, and bounded sequence $(V_k)\subset L^\infty(\Omega)$ we prove the…

In this paper we consider nodal radial solutions $u_\epsilon$ to the problem \[ \begin{cases} -\Delta u=\lambda ue^{u^2+|u|^{1+\epsilon}}&\text{ in }B,\\ u=0&\text{ on }\partial B. \end{cases} \] and we study their asymptotic behaviour as…

We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution $v\in L^\infty (-1,0; L^2 ( B(x_0,r)))\cap L^\infty_{\rm loc} (-1,0; W^{1, \infty}…

In this paper we study the asymptotic behavior of sequences of stationary weak solutions to the following Liouville-type equation $-\Delta u=e^u~~~{in }~~~\Omega$, where $\Omega$ is an open set of $R^3$. By improving the partial regularity…

In this article we are concerned with the existence of blow-up solutions to the following boundary value problem $$-\Delta v= \lambda V(x) |x|^2e^v\;\mbox{in}\quad B_1,\quad v=0 \;\mbox{ on }\quad \partial B_1,$$ where $B_1$ is the unit…

In this paper we study the Euler-Poincar\'{e} equations in $\Bbb R^N$. We prove local existence of weak solutions in $W^{2,p}(\Bbb R^N),$ $p>N$, and local existence of unique classical solutions in $H^k (\Bbb R^N)$, $k>N/2+3$, as well as a…

We study the blow-up behavior of solutions to the singular Liouville equation \[ \Delta \tilde u+\lambda e^{\tilde u}=4\pi\alpha\delta_0 \quad\text{in }B,\quad \tilde u=0 \quad\text{on }\partial B, \] where $\alpha>0$, $\lambda>0$ and…

We construct multiple solutions to the nonlocal Liouville equation \begin{equation} \label{eqk} \tag{L} (-\Delta)^{\frac{1}{2}} u = K(x) e^u \quad \mbox{ in } \mathbb{R}. \end{equation} More precisely, for $K$ of the form $K(x) =…

In this paper we consider the following biharmonic equation with critical exponent $P_\epsilon$ : $\Delta^2 u= Ku^{(n+4)/(n-4)-\epsilon}, u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a domain in $R^n$, $n\geq…

We investigate finite-time blow-up for nonnegative solutions to the Cauchy problem associated with semilinear parabolic equations driven by a mixed local--nonlocal operator. The reaction term is assumed to satisfy suitable structural…

We study blowup solutions of the 6D energy critical heat equation $u_t=\Delta u+|u|^{p-1}u$ in $\R^n\times(0,T)$. A goal of this paper is to show the existence of type II blowup solutions predicted by Filippas, Herrero and Vel\'azquez…

We investigate the blow-up behavior and Liouville-type theorems of solutions to a class of generalized Camassa-Holm-Kadomtsev-Petviashvili (CH-KP) equations with a generally smooth nonlinear term $g(u)$. First, using the continuation…

We study the following Liouville system defined on a compact Riemann surface $M$, \begin{equation} -\Delta u_i=\sum_{j=1}^n a_{ij}\rho_j\Big(\frac{h_j e^{u_j}}{\int_\Omega h_j e^{u_j}}-1\Big)\mbox{ in }M\mbox{ for }i=1,\cdots,n,\nonumber…

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by…

In this paper, we consider the defocusing nonlinear wave equation $-\partial_t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb R\times \mathbb R^d$. Building on our companion work ({\it \small Self-similar imploding solutions of the relativistic Euler…