On isolated singular solutions of semilinear Helmholtz equation.

Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation $$ -\Delta u-u=Q|u|^{p-2}u \quad{\rm in}\ \ \mathbb{R}^N\setminus\{0\},\ \qquad\lim_{|x|\to0}u(x)=+\infty, $$ where $N>2$, $p>1$ and the potential $Q:\mathbb{R}^N\to (0,+\infty)$ is a H\"older continuous function satisfying extra decaying condition at infinity. The isolated singular solutions $u_k=k\Phi+v_k$ is derived by the Schauder fixed point theorem for the integral equation $$v_k=\Phi\ast\big(Q|kw_\sigma+v_k|^{p-2}(kw_\sigma+v_k)\big)\quad{\rm in}\ \, \mathbb{R}^N,$$ where $\Phi$ is the real valued fundamental solution $-\Delta-1$ and $w_\simga$ is weak solution of $-\Delta u-u=\delta_0$ with the behavior at infinity controlled by $|x|^{-\sigma}$.