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

Article type: Research Article

Authors: Aghajani, A.^{; *} | Mottaghi, S.F.

Affiliations: School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran. E-mails: aghajani@iust.ac.ir, mottaghi@mathdep.iust.ac.ir

Correspondence: [*] Corresponding author. E-mail: aghajani@iust.ac.ir.

Abstract: In this paper we consider the p-Laplace equation −Δpu=λf(u) in a smooth bounded domain Ω⊂RN with zero Dirichlet boundary condition, where p>1, λ>0 and f:[0,∞)→R is a C1 function with f(0)>0, f′⩾0 and limt→∞f(t)tp−1=∞. For the sequence (uλ)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(uλ)‖L1(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+4p/(p−1), which is conjectured to be the optimal regularity dimension for u∗.

Keywords: p-Laplacian, semi-stability, extremal solution, regularity, asymptotic behavior

DOI: 10.3233/ASY-201613

Journal: Asymptotic Analysis, vol. 122, no. 1-2, pp. 119-130, 2021