Singular mean field equations on compact Riemann surfaces

For a general class of elliptic PDE's in mean field form on compact Riemann surfaces with exponential nonlinearity, we address the question of the existence of solutions with concentrated nonlinear term, which, in view of the applications, are physically of definite interest. In the model, we also include the possible presence of singular sources in the form of Dirac masses, which makes the problem more degenerate and difficult to attack

Isolated singularities for the n-Liouville equation

In dimension n isolated singularities -- at a finite point or at infinity -- for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in arXiv:1609.03608 and establish a quantization result for entire solutions of the singular n-Liouville equation.Comment: 10 pages; a mistake has been fixe

Sign-Changing Solutions for Critical Equations with Hardy Potential

We consider the following perturbed critical Dirichlet problem involving the Hardy-Schr\"odinger operator on a smooth bounded domain $\Omega \subset \mathbb{R}^N$, $N\geq 3$, with $0 \in \Omega$: $$\left\{ \begin{array}{ll}-\Delta u-\gamma \frac{u}{|x|^2}-\epsilon u=|u|^{\frac{4}{N-2}}u &\hbox{in }\Omega u=0 & \hbox{on }\partial \Omega, \end{array}\right.$$ when $\epsilon>0$ is small and $\gamma< {(N-2)^2\over4}$. Setting $\gamma_j= \frac{(N-2)^2}{4}\left(1-\frac{j(N-2+j)}{N-1}\right)\in(-\infty,0]$ for $j \in \mathbb{N},$ we show that if $\gamma\leq \frac{(N-2)^2}{4}-1$ and $\gamma \neq \gamma_j$ for any $j$, then for small $\epsilon$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $\gamma<\frac{(N-2)^2}{4}-4,$ then for any integer $k \geq 2$, the equation has for small enough $\epsilon$, a sign-changing solution that develops into a superposition of $k$ bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition that $\gamma\neq \gamma_j$ is not necessary. Indeed, it is known that, if $\gamma > \frac{(N-2)^2}{4}-1$ and $\Omega$ is a ball $B$, then there is no radial positive solution for $\epsilon>0$ small. We complete the picture here by showing that, if $\gamma\geq \frac{(N-2)^2}{4}-4$, then the above problem has no radial sign-changing solutions for $\epsilon>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $\gamma=0$.Comment: 41 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif

Non-topological condensates for the self-dual Chern-Simons-Higgs model

For the abelian self-dual Chern-Simons-Higgs model we address existence issues of periodic vortex configurations -- the so-called condensates-- of non-topological type as $k \to 0$, where $k>0$ is the Chern-Simons parameter. We provide a positive answer to the long-standing problem on the existence of non-topological condensates with magnetic field concentrated at some of the vortex points (as a sum of Dirac measures) as $k \to 0$, a question which is of definite physical interest.Comment: accepted on Comm. Pure Appl. Mat

Uniqueness of solutions for an elliptic equation modeling MEMS

We study the effect of the parameter $\lambda$, the dimension $N$, the profile $f$ and the geometry of the domain $\Omega \subset\mathbb{R}^N$, on the question of uniqueness of the solutions to the following elliptic boundary value problem with a singular nonlinearity: 180pt {{array}{ll} -\Delta u= \frac{\lambda f(x)}{(1-u)^2} & \hbox{in}\Omega 0 This equation has been proposed as a model for a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 below a rigid ground plate located at height z = 1.Comment: 11 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.birs.ca/~nassif