Some results on standing wave solutions for a class of quasilinear Schrödinger equations

In this paper, we study the following quasilinear Schrödinger equations -Δu+V(x)u+κ2Δ(u2)u=f(u)+μ|u|2-2u, xRN, where N ≥ 3, κ > 0, μ ≥ 0, and V:RN→R satisfy suitable assumptions. First, by using a change of variable and some new skills, we obtain the ground states for this problem with subcritical growth via the Pohozaev manifold. Second, we establish the existence of ground state solutions with critical growth via Lestimates, which use the method developed by Brezis and Nirenberg [Commun. Pure Appl. Math. 36, 437-477 (1983)] and Jeanjean [Proc. R. Soc. Edinburgh, Sect A. 129, 787-809 (1999)]. Moreover, we give the nonexistence of positive solutions for this problem, where the nonlinear term allow general asymptotically linear growth. Our results extend and supplement the results obtained by Severo et al. [J. Differ. Equations 263, 3550-3580 (2017)], Xu and Chen [J. Differ. Equations 265, 4417-4441 (2018)], and Lehrer and Maia [J. Funct. Anal. 266, 213-246 (2014)] and some other related literature.