Estimates of the extremal solution for the bilaplacian with general nonlinearity

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:443

Article

Angel Navarro, Miguel¹  Villegas, Salvador¹ 

[1] Univ Granada, Dept Anal Matemat, E-18071 Granada, Spain

关键词: Biharmonic;    Extremal solution;    Radial;    Stable;   

DOI  :  10.1016/j.jmaa.2016.05.032

来源: Elsevier

【 摘 要 】

Let lambda* > 0 denote the supremum possible value of lambda such that {Delta(2)u = lambda f (u) in B-1, u = partial derivative n/partial derivative n = 0 on partial derivative B-1} has a classical solution, where B-1 is the unit ball in R-N, n is the exterior unit normal vector, and f is an element of C-1(R) is nondecreasing and satisfies f (0) > 0 and f(t)/t -> +infinity as t -> +infinity. For lambda = lambda* this problem possesses a weak solution u*, the so-called extremal solution. We establish the regularity of this extremal solution for N <= 10. For N >= 11 we establish that lim(r -> 0) r(N-8/2) (u*)'(r) = lim(r -> 0) r(N-10/2) u*(r) = 0 for N <= 19 and lim(r -> 0) r(N-9/2) (u*)'(r) r = lim(r -> 0) r(N-11/2) u* (r) = 0 for N >= 20. Our regularity results do not depend on the specific nonlinearity f. (c) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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