New multiple positive solutions for elliptic equations with singularity and critical growth

Abstract

In this note, the existence of multiple positive solutions is established for a semilinear elliptic equation $ -\Delta u=\frac{\lambda}{u^\gamma}+u^{2^*-1},~x\in\Omega,~u=0, x\in\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ ($N\geq3$), $2^*=\frac{2N}{N-2}$, $\gamma\in(0,1)$ and $\lambda>0$ 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 $\Omega$.

Keywords

• semilinear elliptic equations
• critical growth
• singularity
• positive solution