LAUSR

Abstract We consider stable solutions to the equation − Δ p u = f ( u ) in a smooth bounded domain Ω ⊂ R n for a C 1 nonlinearity f. Either in the radial case, or for some model nonlinearities f in a general domain, stable solutions are known to be bounded in the optimal dimension range n p + 4 p / ( p − 1 ) . In this article, under a new condition on n and p, we establish an L ∞ a priori estimate for stable solutions which holds for every f ∈ 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 ∞ bound for stable solutions holding for all C 1 nonlinearities when n p + 4 p / ( 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 .