科研之友
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摘要
We consider the semilinear elliptic equation -Lu = f(u) in a general smooth bounded domain Omega subset of R-n with zero Dirichlet boundary condition, where L is a uniformly elliptic operator and f is a C (2) positive, nondecreasing and convex function in [0, infinity) such that f(t)/t -> infinity as t -> infinity. We prove that if u is a positive semistable solution then for every 0 <= beta < 1 we have parallel to f(u) integral(u)(0) f(t) f ''(t) e(2 beta) (integral 0t root f ''(s)/f(s)ds) dt parallel to(L1(Omega)) <= C-beta < infinity, where C-beta is a constant independent of u. As we shall see, a large number of results in the literature concerning a priori bounds are immediate consequences of this estimate. In particular, among other results, we establish a priori L-infinity bound in dimensions n <= 9, under the extra assumption that lim sup(t ->infinity) f(t)f ''(t)/f'(t)(2) < 2/9-2 root 14 congruent to 1.318. Also, we establish a priori L-infinity bound when n <= 5 under the very weak assumption that, for some epsilon > 0, lim inf(t ->infinity) (tf(t))(2-epsilon)/f'(t) > 0 or lim inf(t ->infinity) t(2)f(t)f ''(t)/f'(t)(3/2+epsilon) > 0.
- 出版日期2016-5
