A Positive Solution for a Weighted Anisotropic p-Laplace Equation Involving Vanishing Potential

Here, a weighted anisotropic p-Laplace equation \(\begin{aligned} -\Delta _{a\overrightarrow{p}}u+V(x)|x|^{-ap^*}|u|^{p^+-2}u=f(u), \end{aligned}\) $$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{a\overrightarrow{p}}u+{V(x)|x|}^{-a{p}^{\ast }}{|u|}^{p^{+}-2}u=f(u),\end{array}$$ in \(\mathbb {R}^N\) ${\mathbb{R}}^N$ is considered where \(\Delta _{a\overrightarrow{p}}u:=\sum _{i=1}^N\frac{\partial }{\partial x_i}\left( |x|^{-ap_i} \left| \frac{\partial u}{\partial x_i}\right| ^{p_i-2}\frac{\partial u}{\partial x_i}\right) \) ${\mathrm{\Delta }}_{a\overrightarrow{p}}u:={\sum }_{i=1}^N\frac{\partial }{\partial x_i}\left({|x|}^{-a{p}_i},{\left|\frac{\partial u}{\partial x_i}\right|}^{p_i-2},\frac{\partial u}{\partial x_i}\right)$ , the potential V can vanish at infinity with exponential decay, f is a function with subcritical growth of class \(C^1\) $C^1$ , \(\overrightarrow{p}=(p_1,\dots , p_N)\) $\overrightarrow{p}=(p_1,\cdots ,p_N)$ , \(p^+=\max _{1\le i\le N}p_i\) $p^{+}={max}_{1\le i\le N}{p}_i$ and \(p^*=\frac{Np^+}{N-p^+}\) $p^{\ast }=\frac{N{p}^{+}}{N-p^{+}}$ . Because of the lack of compactness, it is necessary to apply Del Pino and Felmer’s arguments and the Moser iteration method to study the existence and properties of a positive solution.