Local and global solutions for some parabolic nonlocal problems

We study local and global existence of solutions for some semilinear parabolic initial boundary value problems with autonomous nonlinearities having a "Newtonian" nonlocal term

Exact Morse index computation for nodal radial solutions of Lane-Emden problems

We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }B u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $B$ is the unit ball of $\mathbb R^N$, $N\geq2$, centered at the origin and $1<p<p_S$, with $p_S=+\infty$ if $N=2$ and $p_S=\frac{N+2}{N-2}$ if $N\geq3$. Our main result is to prove that in dimension $N=2$ the Morse index of the least energy sign-changing radial solution $u_p$ of \eqref{problemAbstract} is exactly $12$ if $p$ is sufficiently large. As an intermediate step we compute explicitly the first eigenvalue of a limit weighted problem in $\mathbb R^N$ in any dimension $N\geq2$

Asymptotic analysis and sign changing bubble towers for Lane-Emden problems

We consider the semilinear Lane-Emden problem in a smooth bounded domain of the plane. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions as the exponent p of the nonlinearity goes to infinity. Among other results we show, under some symmetry assumptions on the domain, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as p goes to infinity, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville problem in the plane

Asymptotic profile of positive solutions of Lane-Emden problems in dimension two

We consider families $u_p$ of solutions to the problem \begin{equation}\label{problemAbstract} \left\{\begin{array}{lr}-\Delta u= u^p & \mbox{ in }\Omega\\ u>0 & \mbox{ in }\Omega\\ u=0 & \mbox{ on }\partial \Omega \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $p>1$ and $\Omega$ is a smooth bounded domain of $\mathbb R^2$. We give a complete description of the asymptotic behavior of $u_p$ as $p\rightarrow +\infty$, under the condition \[p\int_{\Omega} |\nabla u_p|^2\,dx\rightarrow \beta\in\mathbb R\qquad\mbox{ as $p\rightarrow +\infty$}.\

Sign changing solutions of Lane Emden problems with interior nodal line and semilinear heat equations

We consider the semilinear Lane Emden problem in a smooth bounded simply connected domain in the plane, invariant by the action of a finite symmetry group G. We show that if the orbit of each point in the domain, under the action of the group G, has cardinality greater than or equal to four then, for p sufficiently large, there exists a sign changing solution of the problem with two nodal regions whose nodal line does not touch the boundary of the domain. This result is proved as a consequence of an analogous result for the associated parabolic problem