194 research outputs found
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 , , with : when is small and .
Setting for we show that if and for any , then for small , the above equation has a
positive --non variational-- solution that develops a bubble at the origin. If
moreover then for any integer , the
equation has for small enough , a sign-changing solution that
develops into a superposition of bubbles with alternating sign centered at
the origin. The above result is optimal in the radial case, where the condition
that is not necessary. Indeed, it is known that, if
and is a ball , then there is no
radial positive solution for small. We complete the picture here
by showing that, if , then the above problem
has no radial sign-changing solutions for small. These results
recover and improve what is known in the non-singular case, i.e., when
.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 , where 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 , 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 , the dimension , the
profile and the geometry of the domain , 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
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