  We are looking for solutions to nonlinear Schr\"odinger-type equations of the
form $$ (-\Delta)^{\alpha / 2} u (x) + V(x) u(x) = h (x,u(x)), \quad x \in
\mathbb{R}^N, $$ where $V : \mathbb{R}^N \rightarrow \mathbb{R}$ is an external
potential (bounded, unbounded or singular) and the nonlinearity $h :
\mathbb{R}^N \times \mathbb{R} \rightarrow \mathbb{R}$ may be sign-changing.
The aim of this paper is to give an introduction to the problem and the
variational approach based on the Nehari manifold technique, and to survey
recent results concerning the existence of ground and bound states.
