  This paper is dedicated to studying the nonlinear Schr\"odinger equations of
the form \begin{equation*}\label{KE}
  \left\{
  \begin{array}{ll}
  -\triangle u+V(x)u=f(u), & x\in \R^N;
  u\in H^1(\R^N),
  \end{array}
  \right. \end{equation*} where $V\in \mathcal{C}^1(\R^N, [0, \infty))$
satisfies some weak assumptions, and $f\in \mathcal{C}(\R, \R)$ satisfies the
general Berestycki-Lions assumptions. By introducing some new tricks, we prove
that the above problem admits a ground state solution of Poho\u{z}aev type and
a least energy solution. These results generalize and improve some ones in [L.
Jeanjean, K. Tanka,Indiana Univ. Math. J. 54 (2005), 443-464], [L. Jeanjean, K.
Tanka, Proc. Amer. Math. Soc. 131 (2003) 2399-2408], [H. Berestycki, P.L.
Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related
literature. In particular, our assumptions are "almost" necessary when
$V(x)\equiv V_{\infty}>0$, moreover, our approach could be useful for the study
of other problems where radial symmetry of bounded sequence either fails or is
not readily available, or where the ground state solutions of the problem at
infinity are not sign definite.
