Remarks on the Berestycki–Lions Conditions for the Existence of Solutions

In this paper, we are concerned with the following Schrödinger equation \( - \Delta u = g(u),\,\,\,\,\,\,x \in {\mathbb{R}^N}.\) $$-\Delta u=g(u),x\in {ℝ}^N.$$ We give a new approach and a brief proof to show the existence of infinitely many solutions and ground state solutions with g satisfying the Berestycki–Lions conditions [Arch. Rational Mech. Anal., 1983, 82(4): 313–345, 347–357].