Sign-changing solutions of Lane Emden problems with interior nodal line and semilinear heat equations

We consider the semilinear Lane Emden problem{−Δu=|u|p−1uin Ω,u=0on ∂Ω where Ω   is a smooth bounded simply connected domain in R2R2, invariant by the action of a finite symmetry group G.We show that if the orbit of each point in Ω, under the action of the group G, has cardinality greater than or equal to 4 then, for p   sufficiently large, there exists a sign-changing solution of (EpEp) with two nodal regions whose nodal line does not touch ∂Ω.This result is proved as a consequence of an analogous result for the associated parabolic problem.