Multiplicity of singular solutions for semilinear elliptic equations with superlinear source terms

TL;DR

This paper explores the multiple singular solutions of certain nonlinear elliptic equations near the origin, extending known results to a broad class of nonlinear functions using classification and transformation techniques.

Contribution

It generalizes multiplicity results for singular solutions from specific power nonlinearities to a wide range of complex nonlinear functions.

Findings

Multiple singular solutions exist for various nonlinearities near the origin.

Results extend known multiplicity theorems to more general nonlinear functions.

The methods apply to nonlinearities involving powers, logarithms, and exponential functions.

Abstract

This paper investigates the multiplicity of singular solutions for the nonlinear elliptic equation $-\Delta u =f(u)$ near the origin. Applying the classification of nonlinear functions and the transformation, which were developed by the authors, we generalize the multiplicity results known for the concrete model nonlinearity $f(u)=u^p$ with $\frac{N}{N-2}<p<\frac{N+2}{N-2}$. Our result applies to various nonlinearities, such as $f(s)=s^p+s^r$ with $0<r<p$, $f(s)=s^p(\log s)^r$ with $r\in \mathbb{R}$, $f(s)=s^p\exp((\log s)^r)$ with $0<r<1$ and $f(s)=s^p+s^r(\log s)^{\beta}$ with $0<r<p$ and $\beta \in \mathbb{R}$, for $\frac{N}{N-2}<p<\frac{N+2}{N-2}$.