The Ljapunov-Schmidt reduction for some critical problems

This is a survey about the application of the Ljapunov-Schmidt reduction for some critical problems

Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter $\gamma\in(0,1]$ corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for $\gamma=1$. Thus, by analyzing the case $\gamma\neq1$ we emphasize specific properties of the physically relevant parameter $\gamma$ in the vortex concentration phenomena

Large mass boundary condensation patterns in the stationary Keller–Segel system

We consider the boundary value problem {−Δu+u=λeu,inΩ∂νu=0on∂Ω where Ω is a bounded smooth domain in R2, λ>0 and ν is the inner normal derivative at ∂Ω. This problem is equivalent to the stationary Keller–Segel system from chemotaxis. We establish the existence of a solution uλ which exhibits a sharp boundary layer along the entire boundary ∂Ω as λ→0. These solutions have large mass in the sense that ∫Ωλeujavax.xml.bind.JAXBElement@7ff7a637∼|log⁡λ|