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# 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 Salón de seminarios Graciela Salicrup vCal iCal

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.