Mathematik A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains by Joe McKenna
Mathematik A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains by Joe McKenna
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数学 半线性方程的先验界限和 Lipschitz 域的一类新的临界指数作者：Joe McKenna
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
W. Reichel
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作者：
M. Ahmedou;W. Reichel
研究了在有界区域Ω <$R上的半线性椭圆边值问题-<$u = f（x，u），其中u = 0在<$Ω上，非线性0 ≤ f（x，s）至多像s一样增长的正的很弱解的先验界.如果Ω是Lipschitz域，我们给出了两个指数p和p，它们依赖于Green函数的边界行为和Ω的最小内开角。证明了当1 < p < p < p <1时，所有正的很弱解在L∞中是先验有界的.当p > p ∞时，我们构造了一个非线性函数f（x，s）= a（x）s，并给出了一个不属于L∞的正的很弱解.最后，我们给出了一类p ∈ P ∈ N的整环.对于这样的区域，我们已经找到了一个真正的临界指数非常弱的解决方案。在光滑域的情况下，p *= p *= n+1 n-1是一个指数，这在Brezis-Turner [3]的经典工作和Quittner-Soulet [12]的最近工作中是众所周知的。
A-priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems −∆u = f(x, u) on a bounded domain Ω ⊂ R with u = 0 on ∂Ω are studied, where the nonlinearity 0 ≤ f(x, s) grows at most like s. If Ω is a Lipschitz domain we exhibit two exponents p∗ and p∗, which depend on the boundary behaviour of the Green-function and on the smallest interior opening angle of ∂Ω. We prove that for 1 < p < p∗ all positive very weak solutions are a-priori bounded in L∞. For p > p∗ we construct a nonlinearity f(x, s) = a(x)s together with a positive very weak solution which does not belong to L∞. Finally we exhibit a class of domains for which p∗ = p∗. For such domains we have found a true critical exponent for very weak solutions. In the case of smooth domains p∗ = p∗ = n+1 n−1 is an exponent which is well known from classical work of Brezis-Turner [3] and from recent work of Quittner-Souplet [12].