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

Abstract

In this paper we consider the p-Laplace equation − Δ p u = λ f ( u ) in a smooth bounded domain Ω ⊂ R N with zero Dirichlet boundary condition, where p > 1, λ > 0 and f : [ 0 , ∞ ) → R is a C 1 function with f ( 0 ) > 0, f ′ ⩾ 0 and lim t → ∞ f ( t ) t p − 1 = ∞. For the sequence ( u λ ) 0 < λ < λ ∗ 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 ‖ E ( u λ ) ‖ L 1 ( Ω ) is uniformly bounded for λ < λ ∗ . Then using elliptic regularity theory we provide some new L ∞ 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 ∗ ∈ L ∞ ( Ω ) for N < p + 4 p / ( p − 1 ), which is conjectured to be the optimal regularity dimension for u ∗ .