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

We investigate the singular solutions for the nonlinear elliptic equation -Δ u =f(u) near the origin, and generalize the known multiplicity results for the model nonlinearity f(u)=uᵖ with N/N-2<p<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)=sᵖ+sʳ with $$0<r<p$$ , f(s)=sᵖ(log s)ʳ with r∈ R , f(s)=sᵖexp ((log s)ʳ) with $$0<r<1$$ and f(s)=sᵖ+sʳ(log s)β with $$0<r<p$$ and β ∈ R , for N/N-2<p<N+2/N-2 .