Quasilinear elliptic equations involving critical Sobolev exponents
Quasilinear elliptic equations involving critical Sobolev exponents
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DOI：
10.1016/0362-546x(89)90020-5
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发表时间：
1989-08
影响因子：
1.4
通讯作者：
M. Guedda;L. Véron
中科院分区：
文献类型：
--
作者：
M. Guedda;L. Véron
设G是lRN和1< p< n的有界开子集，本文的主要目的是研究拟线性方程i-div (] Du] p- zdu)= a (x) u+ up*- ' in G u> o in G(0.1) u= o在aG上的解u的存在性，其中QEL " (G)和p*= Np/(N-p)。（0.1）的解对应于定义在W上的以下泛函的临界点，，‘ * p (G)Asp*是WdVp (G)在tp ’ (G)中的非紧嵌入所对应的临界Sobolev指数，Q一般不满足Palais-Smale条件，通过简单的变分论证不可能得到Q的临界点。当p= 2时，方程（0.1）已被深入研究。在这种情况下，第一个引人注目的结果是Pohozaev[21]，他证明了如果G在某一点上是星形的，并且u= 0，那么（0.1）不允许解，但是当a不为零时，Brezis和Nirenberg证明了[5]，情况是不同的：当n24方程
LET G be a bounded open subset of lRN and 1< p< N. The main goal of this work is to study the existence of a solution u to the following quasilinear equation i-div (] Du] p-ZDU)= a (x) u+ up*-’ in G u> o in G(0.1) u= o on aG where QEL”(G) and p*= Np/(N-p). The solutions of (0.1) correspond to the critical points of the following functional defined on W,,‘* p (G)Asp* is the critical Sobolev exponent corresponding to the noncompact embedding of WdVp (G) into tp’(G), Q does not in general satisfy the Palais-Smale condition and it is not possible to obtain critical points of Q, via simple variational arguments. When p= 2 equation (0.1) has been intensively studied. The first striking result in that case is due to Pohozaev [21] who proved that if G is starshaped with respect to some point and u= 0 then (0.1) admits no solution, but when a is not zero, Brezis and Nirenberg proved [5] that the situation is different: when N 2 4 the equation