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

TL;DR

This paper proves the existence of sign-changing solutions with interior nodal lines for semilinear Lane Emden problems in symmetric domains, using results from associated semilinear heat equations for large p.

Contribution

It establishes the existence of interior nodal solutions with specific symmetry properties for large p, linking elliptic and parabolic problem analyses.

Findings

Existence of sign-changing solutions with interior nodal lines for large p

Solutions have two nodal regions and do not touch the boundary

Results depend on symmetry group properties of the domain

Abstract

We consider the semilinear Lane Emden problem in a smooth bounded simply connected domain in the plane, invariant by the action of a finite symmetry group G. We show that if the orbit of each point in the domain, under the action of the group G, has cardinality greater than or equal to four then, for p sufficiently large, there exists a sign changing solution of the problem with two nodal regions whose nodal line does not touch the boundary of the domain. This result is proved as a consequence of an analogous result for the associated parabolic problem.