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

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 $p\rightarrow +\infty$}.\

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

Aggregates relaxation in a jamming colloidal suspension after shear cessation

The reversible aggregates formation in a shear thickening, concentrated colloidal suspension is investigated through speckle visibility spectroscopy, a dynamic light scattering technique recently introduced [P.K. Dixon and D.J. Durian, Phys. Rev. Lett. 90, 184302 (2003)]. Formation of particles aggregates is observed in the jamming regime, and their relaxation after shear cessation is monitored as a function of the applied shear stress. The aggregates relaxation time increases when a larger stress is applied. Several phenomena have been proposed to interpret this behavior: an increase of the aggregates size and volume fraction, or a closer packing of the particles in the aggregates.Comment: 7 pages, 7 figures; added figures included in the pdf versio

Blow up of solutions of semilinear heat equations in non radial domains of R2\mathbb R^2

We consider the semilinear heat equation \begin{equation}\label{problemAbstract}\left\{\begin{array}{ll}v_t-\Delta v= |v|^{p-1}v & \mbox{in}\Omega\times (0,T)\\ v=0 & \mbox{on}\partial \Omega\times (0,T)\\ v(0)=v_0 & \mbox{in}\Omega \end{array}\right.\tag{$\mathcal P_p$} \end{equation} where $p>1$, $\Omega$ is a smooth bounded domain of $\mathbb R^2$, $T\in (0,+\infty]$ and $v_0$ belongs to a suitable space. We give general conditions for a family $u_p$ of sign-changing stationary solutions of \eqref{problemAbstract}, under which the solution of \eqref{problemAbstract} with initial value $v_0=\lambda u_p$ blows up in finite time if $|\lambda-1|>0$ is sufficiently small and $p$ is sufficiently large. Since for $\lambda=1$ the solution is global, this shows that, in general, the set of the initial conditions for which the solution is global is not star-shaped with respect to the origin. In previous paper by Dickstein, Pacella and Sciunzi this phenomenon has already been observed in the case when the domain is a ball and the sign changing stationary solution is radially symmetric. Our conditions are more general and we provide examples of stationary solutions $u_p$ which are not radial and exhibit the same behavior