Interior H\"older regularity for stable solutions to semilinear elliptic equations up to dimension 5

Let $2\le n\le 5$. We establish an apriori interior H"older regularity of $C^2$-stable solutions to the semilinear equation $-\mathrm{\Delta }u=f(u)$ in any domain of $R^n$ for any nonlinearity $f\in C^{0,1}(R)$.If $f$ is nondecreasing and convex in addition,we obtain an interior H"older regularity, and hence the local boundedness, of $W^{1,2}(\mathrm{\Omega })$-stable solutions by locally approximating them via $C^2(\mathrm{\Omega })$-stable solutions. In particular, we do not require any lower bound on $f$.