On isolated singular solutions of semilinear Helmholtz equation

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