Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials

Abstract: 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.