New multiple positive solutions for elliptic equations with singularity and critical growth - Hongmin Suo; Chunyu Lei; Jia-Feng Liao (Electronic Journal of Qualitative Theory of Differential Equations, 2019)
In this note, the existence of multiple positive solutions is established for a semilinear elliptic equation $ -Δ u=\frac{λ}{u^γ}+u^{2^*-1},~x\inΩ,~u=0, x\in\partialΩ$, where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$ ($N≥3$), $2^*=\frac{2N}{N-2}$, $γ\in(0,1)$ and $λ>0$ is a real parameter. We show by the variational methods and perturbation functional that the problem has at least two positive solutions $w_0(x)$ and $w_1(x)$ with $w_0(x)<w_1(x)$ in $Ω$.