Infinitely many positive solutions for a class of semilinear elliptic equations

For \begin{document}$ N\geq 3 $\end{document} and \begin{document}$ 1<p<\frac{N+2}{N-2} $\end{document}, we consider the following semilinear elliptic equation

\begin{document}$ \begin{equation*}\label{0-1} \left\{ \begin{aligned} -\Delta u& = V(|x|)(|u-1|^p-1)~~\mathrm{in}~\mathbb{R}^N, \\ u&>0~~\mathrm{in}~\mathbb{R}^N, \quad u\in H^1(\mathbb{R}^N), \end{aligned} \right. ~~~{(1)} \end{equation*}$ \end{document}

where \begin{document}$ V(r) $\end{document} is a positive bounded function satisfying

\begin{document}$ V(r) = 1+\frac{a_1}{r^\alpha}+\frac{a_2}{r^{\alpha+1}}+O\left(\frac{1}{r^{\alpha+1+\theta}}\right)\quad\mathrm{as}\ r\to+\infty. $\end{document}

Here \begin{document}$ a_{1}, \theta>0 $\end{document}, \begin{document}$ a_{2}\in \mathbb{R} $\end{document} and \begin{document}$ \alpha>2(\min\left\{1, (p-1)\right\})^{-1} $\end{document} are some constants. By the finite dimensional Lyapunov-Schmidt reduction method, we show that \begin{document}$ (1) $\end{document} has infinitely many non-radial positive solutions.