Atlas: Nonexistence for some problems of Chipot-Weissler type by Evgeny Galakhov

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Equadiff 2003 - International Conference on Differential Equations
July 22-26, 2003
LUC Diepenbeek
Hasselt, Belgium

Organizers
Freddy Dumortier (Chair, LUC Diepenbeek), Henk Broer (Univ. Groningen), Jean-Pierre Gossez (Univ. Libre Bruxelles), Jean Mawhin (Univ. Cath. Louvain-la-Neuve), Andre Vanderbauwhede (Univ. Gent), Sjoerd Verduyn Lunel (Univ. Leiden)

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Nonexistence for some problems of Chipot-Weissler type
by
Evgeny Galakhov
Universitaet Rostock

Let \Omega be a smooth bounded domain. We consider an elliptic partial differential inequality

\labele1-div(|Ñu|^p-2Ñu) >= \fracu^q|x|^-\alpha-b|Ñu|^s (x in \Omega)

(\theequation)

and evolutional problems with a similar stationary part. It is a generalization of the Chipot-Weissler equation considered in [1].

Adapting for our purposes the methods developed e.g. in [2] (see also the monograph [3]), we establish, in particular, the following result.

Theorem. Let p > 1, q > p-1, \alpha > p, and 0 < s <= \fracpqq+1. Then inequality () has no nontrivial positive solutions in the weak sense, regardless of boundary conditions, for any b > 0.

References

1. M. Chipot and F.B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907.

2. E. Mitidieri and S. Pohozaev, Fujita type theorems for quasilinear parabolic inequalities with gradient nonlinearities, Dokl. Math., Vol. 386 (2002), 160-164.

3. E. Mitidieri and S. Pohozaev, Non existence of positive solution for systems of quasilinear elliptic equations and inequalities in \R^N, Proc. Steklov Inst. Math. 227 (1999).

Date received: May 26, 2003