On isolated singular solutions of semilinear Helmholtz equation

Our purpose of this paper is to study isolated singular solutions semilinear Helmholtz equation \begin{document}$ -\Delta u-u = Q|u|^{p-1}u \quad{\rm in}\ \ \mathbb{R}^N\setminus\{0\},\ \qquad\lim\limits_{|x|\to0}u(x) +\infty, $\end{document} where N\geq 2 $\end{document}, p>1 and the potential Q: \mathbb{R}^N\to (0,+\infty) a Hölder continuous function satisfying extra decaying conditions at infinity. We give classification singularity in Serrin's subcritical case then are derived with form u_k k\Phi+v_k via Schauder fixed point theorem for integral equation$ v_k \Phi\ast\big(Q|kw_\sigma+v_k|^{p-1}(kw_\sigma+v_k)\big)\quad{\rm \, \mathbb{R}^N, $where \Phi real valued fundamental solution -\Delta-1 w_\sigma also (-\Delta-1)w_\sigma \delta_0 asymptotic behavior infinity controlled by |x|^{-\sigma} some \sigma\leq \frac{N-1}{2} $\end{document}.