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∗.

中文翻译：

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具有一般非线性的p-Laplace方程半稳定解的先验估计。

在本文中，我们考虑在Dirichlet边界为零的光滑有界域Ω⊂RN中的p-Laplace方程-Δpu=λf（u），其中p> 1，λ> 0且f：[0，∞）→R为一个C1函数，其中f（0）> 0，f'⩾0且limt→∞f（t）tp-1 =∞。对于最小半稳定解的序列（uλ）0 <λ<λ∗，通过应用半稳定性不等式，我们发现一类函数E渐近地表现为无穷大的f的幂，并证明‖E（uλ ）‖L1（Ω）对于λ<λ∗是有界的。然后，使用椭圆正则性理论，在非线性f的某些合适条件下，为极值解u ∗提供了一些新的L∞估计，其中所获得的结果既不需要f的凸度，也不需要域的严格凸度。特别是，在对f的一些温和假设下，我们证明对于N <p + 4p /（p-1），u ∗∈L∞（Ω），