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# 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 Salón 2 vCal iCal

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).