We will present some recent results on the existence of a positive solution to nonlinear Schrödinger equations $$-\Delta u+V(x)u=f(u)\mbox{ in }\mathbb{R}^N, $$ for very general potentials $V$, with positive or zero limit at infinity (allowing convergence from above, below, or oscillations), but imposing no symmetry or decay rate assumption. Also, the nonlinearities $f$ may satisfy mild hypotheses, including superlinear or asymptotically linear growth at infinity. This is possible due to the application of the classical Monotonicity Trick approach. If time permits, we will also discuss new results for the case where the operator $-\Delta + V(x)$ has negative eigenvalues. This is a work in collaboration with Romildo Lima (UFCG, Brazil) and Mayra Soares (UnB, Brazil)