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In this article, we prove that the least energy nodal solutions to Lane-Emdenequation $-{\Delta}u = |u|^{p-2}u$ with zero Dirichlet boundary conditions on asquare are odd with respect to one diagonal and even with respect to the otherone when $p$ is close to 2. We also show that this symmetry breaks onrectangles close to squares.

A computer assisted proof of the symmetries of least energy nodal solutions on squares