Existence and Asymptotic Behavior of Positive Solutions for a Class of Quasilinear Schrödinger Equations

In this paper, we study the quasilinear Schrödinger equation{-\Delta u+V(x)u-\frac{\gamma}{2}(\Delta u^{2})u=|u|^{p-2}u},{x\in\mathbb{R}^{N}}, where{V(x):\mathbb{R}^{N}\to\mathbb{R}}is a given potential,{\gamma>0}, and either{p\in(2,2^{*})},{2^{*}=\frac{2N}{N-2}}for{N\geq 4}or{p\in(2,4)}for{N=3}. If{\gamma\in(0,\gamma_{0})}for some{\gamma_{0}>0}, we establish the existence of a positive solution{u_{\gamma}}satisfying{\max_{x\in\mathbb{R}^{N}}|\gamma^{\mu}u_{\gamma}(x)|\to 0}as{\gamma\to 0^{+}}for any{\mu>\frac{1}{2}}. Particularly, if{V(x)=\lambda>0}, we prove the existence of a positive classical radial solution{u_{\gamma}}and up to a subsequence,{u_{\gamma}\to u_{0}}in{H^{2}(\mathbb{R}^{N})\cap C^{2}(\mathbb{R}^{N})}as{\gamma\to 0^{+}}, where{u_{0}}is the ground state of the problem{-\Delta u+\lambda u=|u|^{p-2}u},{x\in\mathbb{R}^{N}}.