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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 / Orbits and nodal equilibria in a semilinear parabolic equation with symmetry

Orbits and nodal equilibria in a semilinear parabolic equation with symmetry

Ponente: Fernando García Ruiz
Institución: IM-UNAM
Cuándo 21/05/2015
de 10:00 a 11:00
Dónde Salón de seminarios Graciela Salicrup
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We consider the semilinear elliptic problem
\[
\left\{ \begin{aligned}-\Delta u & =f_{p_{1},p_{2}}\left(u\right), &  & \qquad\mbox{in }\Omega,\\
u & =0, &  & \qquad\mbox{on }\partial\Omega,
\end{aligned}
\right.
\]
where \(\Omega\) is a smooth bounded simply connected domain in \(\mathbb{R}^{2}\),
invariant by the action of a fi{}nite symmetry group \(G\), \(0\in G\),
\(p_{1},p_{2}>1\) and
\[
f_{p_{1},p_{2}}\left(u\right)=\begin{cases}
\left|u\right|^{p_{1}-1}u & \mbox{if }u\geq0\\
\left|u\right|^{p_{2}-1}u & \mbox{if }u<0.
\end{cases}
\]

We show that if the orbit of each point in \(\Omega\), under the action
of the group \(G\), has cardinality greater than or equal to \(4\) then,
for \(p_{1},p_{2}\) sufficiently large and close, there exists a sign
changing solution of the semilinear elliptic problem with two nodal
regions whose nodal line does not touch \(\partial\Omega\).

De Marchis, Ianni and Pacella show the same result for the particular
case with \(p_{1}=p_{2}\). They use strongly that the map \(f_{p_{1},p_{1}}\)
is odd, because they use a topological argument based on the Krasnoselskii
genus. An important point is to replace those arguments.