Nonlinear Schr\"odinger problems: symmetries of some variational solutions

In this paper, we are interested in the nonlinear Schr\"odinger problem $-\Delta u + Vu = \abs{u}^{p-2}u$ submitted to the Dirichlet boundary conditions. We consider $p>2$ and we are working with an open bounded domain $\Omega\subset\IR^N$ ($N\geq 2$). Potential $V$ satisfies $\max(V,0)\in L^{N/2}(\Omega)$ and $\min(V,0)\in L^{+\infty}(\Omega)$. Moreover, $-\Delta + V$ is positive definite and has one and only one principal eigenvalue. When $p\simeq 2$, we prove the uniqueness of the solution once we fix the projection on an eigenspace of $-\Delta + V$. It implies partial symmetries (or symmetry breaking) for ground state and least energy nodal solutions. In the litterature, the case $V\equiv 0$ has already been studied. Here, we generalize the technique at our case by pointing out and explaining differences. To finish, as illustration, we implement the (modified) mountain pass algorithm to work with $V$ negative, piecewise constant or not bounded. It permits us to exhibit direct examples where the solutions break down the symmetries of $V$.