Quasilinear Schrödinger Equations with Stein–Weiss Type Nonlinearity and Potential Vanishing at Infinity

In this paper, we consider the following quasilinear Schrödinger equation involving Stein–Weiss type nonlinearity: \(\begin{aligned}{} & {} -\Delta u+V(x)u-\Delta (u^2)u\\{} & {} \qquad =\frac{1}{|x |^{\alpha }} \left( \int _{\mathbb {R}^N}\frac{G(u(y))}{|y-x |^{\mu }|y |^{\alpha }}dy\right) g(u(x)), \ \ \textrm{in}\ \ \mathbb {R}^N, \end{aligned}\) $$\begin{array}{cc}\hfill & -\mathrm{\Delta }u+V(x)u-\mathrm{\Delta }(u^2)u\hfill \\ \hfill & =\frac{1}{{|x|}^{\alpha }}\left({\int }_{{\mathbb{R}}^N},\frac{G(u(y))}{{|y-x|}^{\mu }{|y|}^{\alpha }},d,y\right)g(u(x)),\text{in}{\mathbb{R}}^N,\hfill \end{array}$$ where \(N\ge 3\) $N\ge 3$ , \(0<\mu <N\) $0<\mu <N$ , \(\alpha \ge 0\) $\alpha \ge 0$ and \(2\alpha +\mu <\min \{\frac{N+2}{2}, 4\}\) $2\alpha +\mu <min\{\frac{N+2}{2},4\}$ and G is the primitive of function g. The potential \(V: \mathbb {R}^N\rightarrow \mathbb {R}\) $V:{\mathbb{R}}^N\to \mathbb{R}$ may decay to zero at infinity. By using variational methods, penalization technique and \(L^{\infty }\) $L^{\infty }$ -estimates, we obtain the existence of a positive solution for the above quasilinear Schrödinger equation.