## Non-existence of positive solutions of Lane-Emden systems.(English)Zbl 0868.35032

Nonexistence of positive solutions of Lane-Emden systems is considered. The system is an extension of the celebrated Lane-Emden equation. The authors point out that:
(1) In the case $$n=3$$, if the positive constants $$p$$ and $$q$$ satisfy $$1/(p+1)+ 1/(q+1)>1/3$$, then the Lane-Emden system does not admit any nonnegative and nontrivial solutions with algebraic growth at infinity.
(2) If $$p$$ and $$q$$ satisfy either $$pq\leq 1$$ or $$pq>1$$ and $$\max\{2(p+1)/(pq-1), 2(q+1)/(pq-1)\}\geq n-2$$, then the Lane-Emden system does not admit any nonnegative and nontrivial solutions.
The proofs depend on asymptotic estimates at infinity. The techniques used in treating the single equation cannot be applied in the case of the system.

### MSC:

 35J60 Nonlinear elliptic equations 35J45 Systems of elliptic equations, general (MSC2000)