Nonexistence of solutions for prescribed mean curvature equations on a ball

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:406

Article

Pan, Hongjing¹  Xing, Ruixiang² 

[1] S China Normal Univ, Sch Math Sci, Guangzhou 510631, Guangdong, Peoples R China

[2] Sun Yat Sen Univ, Sch Math & Computat Sci, Guangzhou 510275, Guangdong, Peoples R China

关键词: Prescribed mean curvature equation;    Radial solution;    Bistable;    Time map;    Quasilinear problem;    Superlinear problem;    Sublinear problem;    Nonexistence;    Nonlinear eigenvalue problem;   

DOI  :  10.1016/j.jmaa.2013.05.003

来源: Elsevier

【 摘 要 】

We prove two nonexistence results of radial solutions to the prescribed mean curvature type problem on a ball {-div(u = 0)(Du/root 1 + vertical bar Du vertical bar(2)) = lambda f(u), x is an element of B-R subset of R-n, x is an element of partial derivative B-R, where lambda is a positive parameter, f is a continuous function with f(0) = 0. Under suitable assumptions on f, we show that the problem with superlinear f has no nontrivial positive solutions for small lambda while the problem with sublinear f has no nontrivial positive solutions for large lambda. The former covers many well-known nonexistence results by Finn, Serrin, Narukawa and Suzuki, Ishimura, Pan and Xing. To the best of the authors' knowledge, the latter is the First nonexistence result involving sublinear mean curvature type equations in higher dimensions. In particular, the sublinear cases contain some important logistic type nonlinearities. These nonexistence results differ greatly from those of semilinear problems. (C) 2013 Elsevier Inc. All rights reserved.

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