科研之友
微信
新浪微博
Facebook
分享链接
摘要
We consider the following semi-linear equations @@@ (-Delta)(p) u = u(+)(gamma) in R-n, @@@ where gamma is an element of (1, n+2p/n-2p), n > 2p > 0, u(+) = max{u, 0}, and 2 <= p is an element of N or p is an element of (0, 1). Subject to the integral constraint @@@ u(+)(gamma) is an element of L-1 (R-n), @@@ we obtain the classification of solutions to the above polyharmonic equation for any gamma < n+2p/n-2p and gamma = n/n-2p, according to the two different assumptions: Delta u(x) -> 0 and u(x) = o(|x|(2)) at infinity, respectively. Under the other integral constraint @@@ u(+)(q) is an element of L-1 (R-n), q = n(gamma - 1)/2p, gamma< n+2p/n-2p, @@@ which is scaling invariant, the classification of solutions with the decay assumption Delta u(x) -> 0 at infinity is established for any integer p >= 2, and the classification of solutions with the growth assumption u(x) = o(|x|(2)) at infinity is proved for integers p = 2, 3 as well. In the fractional equation case, namely p is an element of (0, 1), under either of the above two integral constraints, we also complete the classification of solutions with certain growth assumption at infinity.
