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

TL;DR

This paper proves interior H"older regularity for stable solutions of semilinear elliptic equations in dimensions up to five, without requiring lower bounds on the nonlinearity, extending regularity results for such solutions.

Contribution

It establishes interior regularity for stable solutions to semilinear elliptic equations in low dimensions without lower bounds on the nonlinearity, using approximation techniques.

Findings

Interior H"older regularity for stable solutions in dimensions 2 to 5.

Regularity holds for any nonlinearity in C^{0,1}(R) without lower bounds.

Local boundedness of solutions follows from regularity results.

Abstract

Let $2\le n\le 5$. We establish an apriori interior H\"older 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 H\"older 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 $f$.