We consider the following quasilinear Schrödinger equation
$ - \Delta u + V(x)u - \Delta ({u^2})u = q(x)g(u),\;\;\;\;x \in {\mathbb{R}^N}, $
where $N≥ 1$, $0 < q(x)≤ \lim_{|x|\to∞}q(x)$, $g∈ C(\mathbb{R}^+, \mathbb{R})$ and $g(u)/u^3 \to 1$, as $u \to ∞.$ We establish the existence of a positive solution to this problem by using the method developed in Szulkin and Weth [
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