Isolated Singularities for Semilinear Elliptic Systems with Power-Law Nonlinearity

TL;DR

This paper investigates isolated singularities in semilinear elliptic systems with power-law nonlinearities, revealing new invariants and classifying solutions with detailed asymptotic analysis using advanced mathematical techniques.

Contribution

It introduces a new Pohozaev invariant for systems, classifies singular solutions, and analyzes their asymptotic behavior, extending scalar theory to systems.

Findings

Discovery of a new Pohozaev invariant for systems

Classification of solutions with isolated singularities

Asymptotic behavior analysis of solutions near singularities

Abstract

We study the system $-\Delta \mathbf{u}=| \mathbf{u}|^{\alpha-1} \mathbf{u}$ with $1<\alpha\leq\frac{n+2}{n-2}$, where $ \mathbf{u}=(u_1,\dots,u_m)$, $m\geq 1$, is a $C^2$ nonnegative function that develops an isolated singularity in a domain of $\mathbb{R}^n$, $n\geq 3$. Due to the multiplicity of the components of $ \mathbf{u}$, we observe a new Pohozaev invariant other than the usual one in the scalar case, and also a new class of singular solutions provided that the new invariant is nontrivial. Aligned with the classical theory of the scalar equation, we classify the solutions on the whole space as well as the punctured space, and analyze the exact asymptotic behavior of local solutions around the isolated singularity. On the technical level, we adopt the method of the moving spheres and the balanced-energy-type monotonicity functionals.