574 research outputs found
Multiple blow-up solutions for the Liouville equation with singular data
We study the existence of solutions with multiple concentration to the
following boundary value problem -\Delta u=\e^2 e^u-4\pi \sum_{p\in
Z}\alpha_p \delta_{p}\;\hbox{in} \Omega,\quad u=0 \;\hbox{on}\partial \Omega,
where is a smooth and bounded domain in , 's are
positive numbers, is a finite set, defines the
Dirac mass at , and \e>0 is a small parameter. In particular we extend the
result of Del-Pino-Kowalczyk-Musso (\cite{delkomu}) to the case of several
singular sources. More precisely we prove that, under suitable restrictions on
the weights , a solution exists with a number of blow-up points
up to
A continuum of solutions for the SU(3) Toda System exhibiting partial blow-up
In this paper we consider the so-called Toda System in planar domains under
Dirichlet boundary condition. We show the existence of continua of solutions
for which one component is blowing up at a certain number of points. The proofs
use singular perturbation methods
On the profile of sign changing solutions of an almost critical problem in the ball
We study the existence and the profile of sign-changing solutions to the
slightly subcritical problem -\De u=|u|^{2^*-2-\eps}u \hbox{in} \cB,
\quad u=0 \hbox{on}\partial \cB, where \cB is the unit ball in \rr^N,
, and \eps>0 is a small parameter. Using a
Lyapunov-Schmidt reduction we discover two new non-radial solutions having 3
bubbles with different nodal structures. An interesting feature is that the
solutions are obtained as a local minimum and a local saddle point of a reduced
function, hence they do not have a global min-max description.Comment: 3 figure
Behaviour of symmetric solutions of a nonlinear elliptic field equation in the semi-classical limit: concentration around a circle
In this paper we study the existence of concentrated solutions of the nonlinear field equation where , , , the potential is positive and radial, and is an appropriate singular function satisfying a suitable symmetric property. Provided that is sufficiently small, we are able to find solutions with a certain spherical symmetry which exhibit a concentration behaviour near a circle centered at zero as . Such solutions are obtained as critical points for the associated energy functional; the proofs of the results are variational and the arguments rely on topological tools. Furthermore a penalization-type method is developed for the identification of the desired solutions
Ground state solutions to the nonlinear Schrodinger-Maxwell equations
We prove the existence of ground state solutions for the nonlinear
Schrodinger-Maxwell equations.Comment: 27 page
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