Nonexistence of solutions to systems of higher-order semilinear inequalities in cone-like domains

 

Abdallah El Hamidi and Gennady G. Laptev
Université de la Rochelle
Avenue Michel Crépeau
17042 La Rochelle Cedex 1
France

 

Abstract:

In this paper we obtain nonexistence results of global solutions to the following system of higher-order semilinear partial differential inequalities

$ \left\{ \begin{array}{lll} \frac{\partial^k u_i}{\partial t^k}-\Delta (a_i (x,... ...vert^{p_{i+1}}, \;\;\; 1 \leq i \leq n, \\ \\ u_{n+1}=u_1, \end{array}\right. $

in cones and cone-like domains in $\mathbb{R}^N$, $t>0$. Our results concern as well the nonnegative solutions as the solutions which change sign. Moreover, general formula of the critical exponent corresponding to the previous system is given. Our proofs are based on the test function method, developed by E. Mitidieri and S. I. Pohozaev.

Key words: nonexistence, semilinear inequalities, conical domains, critical exponent.
Mathematics Subject Classification: 35R45 (35G25 35K55 35L70).

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Laboratoire de Mathématiques