  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 $V$.
