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

We consider the semilinear Lane Emden problem {-Delta u = vertical bar u vertical bar(p-1)u in Omega, u = 0 on partial derivative Omega (epsilon(p)) where Omega is a smooth bounded simply connected domain in R-2, invariant by the action of a finite symmetry group G. 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 sufficiently large, there exists a sign-changing solution of (epsilon(p)) with two nodal regions whose nodal line does not touch partial derivative Omega. This result is proved as a consequence of an analogous result for the associated parabolic problem. (C) 2013 Elsevier Inc. All rights reserved.