This article is concerned with the existence of positive ground state solutions for an asymptotically periodic quasilinear Schr\”{o}dinger equation. By using a Nehari-type constraint, we get the existence results which improve the ones in Shi and Chen [Comput. Math. Appl. 71, 2016, 849-858]. Moreover, we give an application of our results, which extends the results in Li [Commun. Pure Appl. Anal. 14, 2015, 1803-1816].

We study the existence of standing wave solutions for the quasilinear Schr\”{o}dinger equation with the critical exponent and singular coefficients. By applying the Mountain Pass Theorem and the Concentration Compactness Principle, we get a ground state solution. Moreover, the asymptotic behavior of the ground state solution is also obtained.

We consider the quasilinear Schr\”{o}dinger equation involving a general nonlinearity at critical growth. By using Jeanjean’s monotonicity trick and the Pohozaev identity, we get the existence results which generalize an earlier work (Liu and Zhao, Commun. Pure Appl. Anal. 19: 3429-3444, 2020) about the subcritical case to the critical case.