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Usted está aquí: Inicio / Actividades / Seminarios / Seminario de Ecuaciones Diferenciales No Lineales (SEDNOL) / Actividades del Seminario de Ecuaciones Diferenciales No Lineales / Entire nodal solutions of a semilinear elliptic equation and their effect on concentration phenomena

Entire nodal solutions of a semilinear elliptic equation and their effect on concentration phenomena

Ponente: Mónica Clapp
Institución: IM-UNAM
Tipo de Evento: Investigación
Cuándo 28/04/2016
de 10:00 a 11:00
Dónde Salón 2
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The aim of this talk is to present some new concentration phenomena for solutions to the problem
\[
\tag{1}\qquad\left\{
\begin{aligned}
-\varepsilon^{2}\Delta u+u&=|u|^{p-2}u && \text{in }\Omega,\\
u&=0 && \text{on }\partial\Omega,
\end{aligned}
\right.
\]
as \(\varepsilon\rightarrow0,\) where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^{N}\), \(N\geq3,\) \(\varepsilon>0\), and \(p\in(2,2^{\ast}),\) with \(2^{\ast}:=\frac{2N}{N-2}\) the critical Sobolev exponent.

This problem appears as a model for pattern formation in various branches of science, e.g., in the study of stationary solutions for the Keller-Segal system in chemotaxis or the Gierer-Meinhardt system in biological pattern formation, and it has been extensibly studied.

A common feature of all available results is that the asymptotic profile of the solutions at the blow-up points is a rescaling of the ground states of the limit problem
\[
\tag{2}\qquad\left\{
\begin{aligned}
-\Delta u+u&=|u|^{p-2}u,\\
u&\in H^{1}(\mathbb{R}^{N}).
\end{aligned}
\right.
\]


We show that there exist a nonradial sign-changing bound state \(\widehat {\omega}\) to the limit problem (2) which has low energy, and sign-changing solutions to (1), which concentrate at a single point, whose asymptotic profile as \(\varepsilon\rightarrow0\) is a rescaling of \(\widehat{\omega}.\)

This is joint work with P.N. Srikanth (Tata Institute of Fundamental Research, Bangalore).