The boundedness of stable solutions to semilinear elliptic equations with linear lower bound on nonlinearities

Let $2\le n\le 9$. Suppose that $f:R\to R$ is locally Lipschitz function satisfying $f(t)\ge A\min \{0,t\}-K$ for all $t\in R$ with some constant $A\ge 0$ and $K\ge 0$. We establish an a priori interior Hölder regularity of $C^2$-stable solution to the semilinear elliptic equation $-\mathrm{\Delta }u=f(u)$. If, in addition, $f$ is nondecreasing and convex, we obtain the interior Hölder regularity of $W^{1,2}$-stable solutions. Note that the dimension $n\le 9$ is optimal.