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

Let 2 ≤ n ≤ 9. Suppose that f : R → R is locally Lipschitz function satisfying f(t) ≥ Amin{0, t} − K for all t ∈ R with some constant A ≥ 0 and K ≥ 0. We establish an a priori interior Hölder regularity of C²-stable solutions to the semilinear elliptic equation −Δu = 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 ≤ 9 is optimal.