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

On the construction of non-simple blow-up solutions for the singular Liouville equation with a potential

We are concerned with the existence of blowing-up solutions to the following boundary value problem −Δu=λV(x)eu−4πNδ0 in B1,u=0 on ∂B1, where B1 is the unit ball in R2 centered at the origin, V(x) is a positive smooth potential, N is a positive integer (N≥1). Here δ0 defines the Dirac measure with pole at 0, and λ&gt;0 is a small parameter. We assume that N=1 and, under some suitable assumptions on the derivatives of the potential V at 0, we find a solution which exhibits a non-simple blow-up profile as λ→0+

Non-symmetric blowing-up solutions for a class of Liouville equations in the ball

We are concerned with the existence of blowing-up solutions to the following boundary value problem: -&amp; UDelta;v = lambda V(x)e(v) in B-1, v = 0 on partial differential B-1, where B-1 is the unit ball in R-2, V(x) is a positive smooth potential, and lambda &gt; 0 is a small parameter. We assume that the potential V satisfies some suitable assumptions in terms of the second and the fourth derivatives at 0, and we find a solution that exhibits a non-symmetric blow-up profile as lambda &amp; RARR; 0(+)