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 $\Omega$ is a smooth and bounded domain in $\R^2$, $\alpha_{p}$'s are positive numbers, $Z\subset \Omega$ is a finite set, $\delta_p$ defines the Dirac mass at $p$, 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 $\alpha_p$, a solution exists with a number of blow-up points $\xi_j\in \Omega\setminus Z$ up to $\sum_{p\in Z}\max\{n\in\N\,|\, n<1+\alpha_p\}$

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, $N\geq 3$, $2^*=\frac{2N}{N-2}$ 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 $$-h^{2}Delta v+V(x)v-h^{p}Delta_{p}v+ W'(v)=0,,$$ where $v:{mathbb R}^{N}o{mathbb R}^{N+1}$, $Ngeq 3$, $p>N$, the potential $V$ is positive and radial, and $W$ is an appropriate singular function satisfying a suitable symmetric property. Provided that $h$ 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 $ho 0^{+}$. 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