具有指数增长的非线性p-双调和问题解的存在性和非存在性

Existence and Nonexistence of Solutions of Nonlinear p-Biharmonic Problem with Exponential Growth

主要考虑了具有指数增长的4阶退化p-双调和问题弱解的存在性和非存在性.当n≥2p+1和2p≥n时,这些结果是不同的.尽管对高阶问题,弱比较原理不再成立,但是仍有其他一些方法可以使用.当2p≥n时,采用的是极小极大方法,而当n≥2p+1时,采用的是上下解方法.最后给出了当n≥2p+1时正则解的渐近行为.

In this paper, a class of fourth-order degenerate elliptic problem involving pbiharmonic operator with exponential growth is investigated in a smooth domain. The existence and nonexistence of solutions in the weak sense are studied. But some results are different between n ≥ 2p ＋ 1 and 2p ≥ n. Although weak comparison principle is not available for high-order problem, other methods still are available. Here, the authors use the minimax method if 2p ≥ n and the method of upper and lower solutions if n ≥ 2p ＋ 1. Finally, the asymptotic behavior of regular solution is established if n ≥ 2p ＋ 1.