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

We investigate the singular solutions for the nonlinear elliptic equation $$-\Delta u =f(u)$$ near the origin, and generalize the known multiplicity results for the model nonlinearity $$f(u)=u^p$$ with $$\frac{N}{N-2}<p<\frac{N+2}{N-2}$$ . Our analysis relies on the transformation introduced in Proposition 3.1 of [Fujishima-Ioku, J. Math. Pures Appl., 118 (2018), 128–158], which enables the reduction of equations with monotonically increasing nonlinearities to the corresponding prototypical power-type case. This approach yields multiplicity results for a broad class of nonlinearities, including $$f(s)=s^p+s^r$$ 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 $$\frac{N}{N-2}<p<\frac{N+2}{N-2}$$ .