The Boundedness of Stable Solutions to Semilinear Elliptic Equations With Linear Lower Bound on Nonlinearities

Abstract

Abstract Let $2\le n\le 9$. Suppose that $f:{{\mathbb {R}}}\to {{\mathbb {R}}}$ is locally Lipschitz function satisfying $f(t)\ge A\min \{0,t\}-K$ for all $t\in {{\mathbb {R}}}$ with some constant $A\ge 0$ and $K\ge 0$. We establish an a priori interior Hölder regularity of $C^{2}$-stable solutions to the semilinear elliptic equation $-\Delta u=f(u)$. 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.