On isolated singular solutions of semilinear Helmholtz equation

TL;DR

This paper classifies isolated singular solutions of the semilinear Helmholtz equation in nd , providing a detailed analysis of their structure and existence using fixed point methods and integral equations.

Contribution

It offers a classification of isolated singularities in the subcritical case and constructs solutions with specific asymptotic behavior using Schauder fixed point theorem.

Findings

Classification of isolated singularities in the subcritical case

Existence of solutions with prescribed asymptotic behavior

Use of integral equations and fixed point theorem for construction

Abstract

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 $\sigma\leq \frac{N-1}{2}$.