Quasilinear elliptic equations with critical exponents
Quasilinear elliptic equations with critical exponents
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DOI：
10.12775/tmna.1996.006
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发表时间：
1996-03
影响因子：
0.7
通讯作者：
P. Clément;D. G. Figueiredo;E. Mitidieri
中科院分区：
文献类型：
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作者：
P. Clément;D. G. Figueiredo;E. Mitidieri
如果Ω⊂R，N≥3关于某点有界且星形，且2∗=2N/(N−2)，则无解。在(P0)中，非线性项是临界指数为(N+2)/(N−2)的u的幂。这一术语源于这样一个事实，即连续SOBLEV嵌入H0(Ω)⊂L(Ω)，对于p≤2∗和Ω有界，除了当p=2∗时，它也是紧的。这种紧致性的损失反映在欧拉-拉格朗日方程为(P0)的泛函不能满足Palais-Smear条件。后来，Brezis和Nirenberg[BN]观察到，帕莱-斯梅尔条件只在某些水平上失效。然后他们证明，如果非线性项稍有扰动，新问题就有解。
has no solution if Ω ⊂ R , N ≥ 3, is bounded and starshaped with respect to some point, and 2∗ = 2N/(N − 2). In (P0) the nonlinear term is a power of u with the critical exponent (N + 2)/(N − 2). This terminology comes from the fact that the continuous Sobolev imbeddings H 0 (Ω) ⊂ L(Ω), for p ≤ 2∗ and Ω bounded, are also compact except when p = 2∗. This loss of compactness reflects in that the functional whose Euler–Lagrange equation is (P0) fails to satisfy the Palais–Smale condition. Later Brezis and Nirenberg [BN] observed that the Palais–Smale condition fails at certain levels only. Then they proved that if the nonlinear term is slightly perturbed, the new problem has a solution.