A quasi-linear elliptic equation with critical growth on compact Riemannian manifold without boundary
A quasi-linear elliptic equation with critical growth on compact Riemannian manifold without boundary
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DOI：
10.1007/s10455-010-9218-0
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发表时间：
2010-07
影响因子：
0.7
通讯作者：
J. Ó;Y. Yang
中科院分区：
文献类型：
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作者：
J. Ó;Y. Yang
设（M，g）是一个N维无边界紧致黎曼流形.当m是严格小于坦恩时，我们证明了其中||u||在亚当斯的原始不等式（Ann Math 128：385-398，1988）中，m，N/mis是通常的Sobolev范数，αN，mis是最佳常数。这是丰塔纳（Comment Math Helv 68：415-454，1993）证明的紧致黎曼流形上的亚当斯不等式的一个修正形式。在m = 1的情形下，利用上述不等式，我们建立了拟线性方程在W_1，N（M）中存在非平凡正弱解的充分条件，其中，和f（x，u）的性质类似于对某个γ> 0.
Let (M,g) be anN-dimensional compact Riemannian manifold without boundary. Whenmis a positive integer strictly smaller thanN, we prove thatwhere ||u||m,N/mis the usual Sobolev norm of, andαN,mis the best constant in Adams’ original inequality (Ann Math 128:385–398, 1988). This is a modified version of Adams’ inequality on compact Riemannian manifold which has been proved by Fontana (Comment Math Helv 68:415–454, 1993). Using the above inequality in the case whenm= 1, we establish sufficient conditions under which the quasi-linear equationhas a nontrivial positive weak solution inW1,N(M), where, andf(x,u) behaves likeasfor someγ> 0.