Boundedness of stable solutions to nonlinear equations involving the p-Laplacian

We consider stable solutions to the equation -Delta(p)u = f(u) in a smooth bounded domain Omega subset of R-n for a C-1 nonlinearity f. Either in the radial case, or for some model nonlinearities fin a general domain, stable solutions are known to be bounded in the optimal dimension range n < p+4p/(p - 1). In this article, under a new condition on nand p, we establish an L-infinity a priori estimate for stable solutions which holds for every f is an element of C-1. Our condition is optimal in the radial case for n >= 3, whereas it is more restrictive in the nonradial case. This work improves the known results in the topic and gives a unified proof for the radial and the nonradial cases. The existence of an L-infinity bound for stable solutions holding for all C-1 nonlinearities when n < p + 4p/(p - 1) has been an open problem over the last twenty years. The forthcoming paper [11] by Cabre, Sanchon, and the author will solve it when p > 2. (C) 2020 Elsevier Inc. All rights reserved.