A priori estimates for semistable solutions of p-Laplace equations with general nonlinearity

In this paper we consider the p-Laplace equation - Delta(p)u = lambda f (u) in a smooth bounded domain Omega subset of R-N with zero Dirichlet boundary condition, where p > 1,. > 0 and f : [0, infinity) -> R is a C-1 function with f (0) > 0, f' >= 0 and lim(t ->infinity) f (t)/t(p-1) = infinity. For the sequence (u.)0<.<.* of minimal semi-stable solutions, by applying the semi-stability inequality we find a class of functions E that asymptotically behave like a power of f at infinity and show that parallel to E(u(lambda))parallel to L-(Omega)(1) is uniformly bounded for. <.*. Then using elliptic regularity theory we provide some new L-infinity estimates for the extremal solution u*, under some suitable conditions on the nonlinearity f, where the obtained results require neither the convexity of f nor the strictly convexity of the domain. In particular, under some mild assumptions on f we show that u* is an element of L-infinity (Omega) for N < p + 4p/( p - 1), which is conjectured to be the optimal regularity dimension for u*.