A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains
A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains
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DOI：
10.1016/j.jfa.2006.11.018
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发表时间：
2007-03
影响因子：
1.7
通讯作者：
P. J. McKenna;W. Reichel
中科院分区：
文献类型：
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作者：
P. J. McKenna;W. Reichel
研究了半线性椭圆型边值问题−Δu=f(x，u)在Ω⊂Rn上具有u=0的有界域上正解、很弱解的先验界，其中非线性项0⩽f(x，S)增长至多如sp.如果Ω是Lipschitz域，我们给出两个指数p*和p*，这取决于格林函数的边界行为和∂Ω的最小内开角。我们证明了对于1<；p<；p*，所有正的很弱解都是先验有界的L∞。对于p>；p*，我们构造了一个非线性函数f(x，S)=a(X)，以及一个不属于L∞的正的很弱解。最后，我们证明了一类p*=p*的域。对于这类区域，我们找到了非常弱解的真实临界指数。在光滑域p*=p*=n+1n的情况下，−1是一个指数，这在Brezis，Turner[H.Brezis，R.E.L.Turner，关于一类超线性椭圆问题Comm.偏微分方程组2(1977)601-614]和Quittner，Souplet[P.Quittner，Ph.Souplet，A先验估计和加权Lebesogue空间中Bootstrap的椭圆组的存在性，Arch.定量配给。机甲。肛门。174(2004)49-81]。
A priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems −Δu=f(x,u) on a bounded domain Ω⊂Rnwith u=0 on ∂Ω are studied, where the nonlinearity 0⩽f(x,s) grows at most like sp. If Ω is a Lipschitz domain we exhibit two exponents p*and p*, which depend on the boundary behavior 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)sptogether 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+1n−1 is an exponent which is well known from classical work of Brezis, Turner [H. Brezis, R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601–614] and from recent work of Quittner, Souplet [P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004) 49–81].