Nonexistence of solutions for prescribed mean curvature equations on a ball

We prove two nonexistence results of radial solutions to the prescribed mean curvature type problem on a ball { − div ( D u 1 + | D u | 2 ) = λ f ( u ) , x ∈ B R ⊆ R n , u = 0 , x ∈ ∂ B R , where λ 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 λ while the problem with “sublinear” f has no nontrivial positive solutions for large λ . 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.