Given a solution \(u\) to the Dirichlet problem \[ \left\{ \begin{aligned} -\Delta u &= f(u), & & \text{in \(\Omega\)},\\ u&=0 & & \text{on \(\partial \Omega\)}, \end{aligned} \right. \] one asks the question under which assumptions the solutions \(u\) does inherit the symmetries of the domain \(\Omega\). Gidas, Ni and Nirenberg have shown that if \(f\) is Lipschitz continuous, \(u\) is positive and \(\Omega\) is a ball, then \(u\) is radial.

With Michel Willem, me have extended a method due to Thomas Bartsch, Michel Willem and Tobias Weth in order to study tge symmetry of least energy nodal solutions of \[ \left\{ \begin{aligned} -\Delta u(x)+a(x)u(x) &= f(x, u(x)), & & \text{in \(x \in \Omega\)},\\ u&=0 & & \text{on \(\partial \Omega\)}, \end{aligned} \right. \] when \(f\) is not Hölder continuous.

With Denis Bonheure, Vincent Bouchez and Christopher Grumiau, we have studied the problem of symmetry of least energy nodal solutions of \[ \left\{ \begin{aligned} -\Delta u &= \lambda u^p, & & \text{in \(\Omega\)},\\ u&=0 & & \text{on \(\partial \Omega\)}. \end{aligned} \right. \] where \(p > 1\). We have studied the asymptotics of the solutions when \(p \to 1\). This lead us to symmetry results when the second eigenvalue of the Laplacian is nondegenerate, some symmetry breaking and a conjecture that when \(\Omega\) is a square, least energy nodal solutions are symmetric with respect to the diagonals.