Abstract :
In this paper we study first nonexistence of radial entire solutions of elliptic systems of the mean curvature
type with a singular or degenerate diffusion depending on the solution u. In particular we extend a
previous result given in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential
Equations 188 (2003) 353–389]. Moreover, in the scalar case we obtain nonexistence of all entire
solutions, radial or not, of differential inequalities involving again operators of the mean curvature type and
a diffusion term.We prove that in the scalar case, nonexistence of entire solutions is due to the explosion of
the derivative of every nonglobal radial solution in the right extremum of the maximal interval of existence,
while in that point the solution is bounded. This behavior is qualitatively different with respect to what
happens for the m-Laplacian operator, studied in [R. Filippucci, Nonexistence of radial entire solutions
of elliptic systems, J. Differential Equations 188 (2003) 353–389], where nonexistence of entire solutions
is due, even in the vectorial case, to the explosion in norm of the solution at a finite point. Our nonexistence
theorems for inequalities extend previous results given by Naito and Usami in [Y. Naito, H. Usami,
Entire solutions of the inequality div(A(|Du|)Du) f (u), Math. Z. 225 (1997) 167–175] and Ghergu
and Radulescu in [M. Ghergu, V. Radulescu, Existence and nonexistence of entire solutions to the logistic
differential equation, Abstr. Appl. Anal. 17 (2003) 995–1003].
© 2007 Elsevier Inc. All rights reserved.
