Semilinear elliptic equations with singular potentials and nonlinearities

This paper is concerned with semilinear elliptic equations $-\Delta u+\frac{\mu }{{\rho }^2}u=\frac{\lambda }{{(a-u)}^2}$ in a $C^2$ bounded convex domain Ω of $R^N$, subject to the zero Dirichlet boundary conditions, where $\mu \in (-\frac{1}{4},0)$ and $\rho \left(x\right)=\mathrm{dist}(x,\partial \Omega )$. We analyze the properties of minimal solutions when $\lambda >0$ and the function $a:\overline{\Omega }\to [0,1]$ satisfying $a\left(x\right)\ge \kappa \rho {\left(x\right)}^{\gamma }$ for some $\kappa >0$ and $\gamma \in (0,\frac{3+\sqrt{1+4\mu }}{4})$. Moreover, the regularity and stability of the extremal solution are obtained.