Interior Hölder regularity for stable solutions to semilinear elliptic equations up to dimension 5

Abstract

Let $2\le n\le 5$. We establish an apriori interior Hölder regularity of $C^2$-stable solutions to the semilinear equation $-\Delta u=f(u)$ 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 Hölder regularity, and hence the local boundedness, of $W^{1,2}(\Omega )$-stable solutions by locally approximating them via $C^2(\Omega )$-stable solutions. In particular, we do not require any lower bound on $f$.