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

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 $\frac{N}{N-2}<p<\frac{N+2}{N-2}$.