Related papers: Bubbling solutions for mean field equations with v…
The understanding of some large energy, negative specific heat states in the Onsager description of 2D turbulence, seems to require the analysis of a subtle open problem about bubbling solutions of the mean field equation. Motivated by this…
In this paper, we extend the analysis of the subcritical approximation of the Nirenberg problem on spheres recently conducted in \cite{MM19, MM}. Specifically, we delve into the scenario where the sequence of blowing up solutions exhibits a…
Let $(M,g)$ be a compact smooth connected Riemannian manifold (without boundary) of dimension $N\ge7$. Assume $M$ is symmetric with respect to a point $\xi_0$ with non-vanishing Weyl's tensor. We consider the linear perturbation of the…
Under the validity of the positive mass theorem, the Yamabe flow on a smooth compact Riemannian manifold of dimension $N \ge 3$ is known to exist for all time $t$ and converges to a solution to the Yamabe problem as $t \to \infty$. We prove…
In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlinear Schr$\ddot{o}$dinger equations on some Riemannian manifolds like the standard 2-sphere $S^2$ and the hyperbolic…
Let $(M,g)$ be a closed locally conformally flat Riemannian manifold of dimension $n \ge 7$ and of positive Yamabe type. If $\xi_0$ denotes a non-degenerate critical point of the mass function we prove the existence, for any $ k \ge 1$ and…
We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval $I$, under Dirichlet conditions in the exterior of $I$. This model is strictly related to the…
We study the existence of multi-bubble solutions for the following skew-symmetric Chern--Simons system \begin{equation}\label{e051} \left\{ \begin{split} &\Delta…
We consider a sinh-Poisson type equation with variable intensities and Dirichlet boundary condition on a pierced domain \begin{equation*} \left\{ \begin{array}{ll} \Delta u +\rho\left(V_1(x)e^{u}- V_2(x)e^{-\tau u}\right)=0 &\text{in }…
We establish the non-degeneracy of bubbling solutions for singular mean field equations when the blow-up points are either regular or involve non-quantized singular sources. This extends the results from Bartolucci-Jevnikar-Lee-Yang…
In this article, we are interested in multi-bump solutions of the singularly perturbed problem \begin{equation*} -\epsilon^2\Delta v+V(x)v=f(v) \ \ \mbox{ in }\ \ \R^N. \end{equation*} Extending previous results \cite{B, DLY,W1}, we prove…
In this paper we study the Palais-Smale sequences of the conformal Dirac-Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the…
In this paper we perform a refined blow up analysis of finite energy approximated solutions to a Nirenberg type problem on half spheres. The later consists of prescribing, under minimal boundary conditions, the scalar curvature to be a…
We obtain existence of double bubbles of large and constant mean curvatures in Riemannian manifolds. These arise as perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned…
The seminal work \cite{bm} by Brezis and Merle showed that the bubbling solutions of the mean field equation have the property of mass concentration. Recently, Lin and Tarantello in \cite{lt} found that the "bubbling implies mass…
We consider the supercritical Lane-Emden problem $$(P_\eps)\qquad -\Delta v= |v|^{p_\eps-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} $$ where $\mathcal A$ is an annulus in $\rr^{2m},$ $m\ge2$ and…
Let $(M,g)$ be a compact Riemann surface with unit area. We investigate the mean field equation for equilibrium turbulence: \begin{align} \begin{cases} -\Delta u = \rho_1\left(\frac{h_1e^{u}}{\int_Mh_1e^udv_g}-1\right) -…
We consider the porous medium equation with power-type reaction terms $u^p$ on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If $p>m$, small data give rise to…
Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -\Delta_{g}u+au=0 & \text{ on }M \\ \partial_\nu u+\frac{n-2}{2}bu= u^{{n\over…
We construct $C^\infty$ solutions to the one-dimensional nonlinear wave equation $$ u_{tt} - u_{xx} - \tfrac{2(p+2)}{p^2} |u|^p u=0 \quad \text{with} \quad p>0 $$ that blow up on any prescribed uniformly space-like $C^\infty$ hypersurface.…