## Nonexistence of positive solutions of semilinear elliptic systems in $$\mathbb{R}^ N$$.(English)Zbl 0848.35034

The author studies nonexistence results for the elliptic system $- \Delta u= f(x, u, v),\quad - \Delta v= g(x, u, v)\quad\text{in} \quad \mathbb{R}^N,$ where $$N\geq 3$$ and $$f$$ and $$g$$ are nonnegative continuous functions. The main results are related to right-hand sides of the following type:
(i) $$f(u, v)\geq a_{11} u^k+ a_{12} v^p$$, $$g(u, v)\geq a_{21} u^q+ a_{22} v^s$$ for $$u, v\geq 0$$ with $$f(0, 0)= g(0, 0)= 0$$ and constants $$a_{ij}> 0$$ and $$k, p, q, s> 1$$.
(ii) $$f(u, v)= u^k v^p$$, $$g(u, v)= u^q v^s$$ with $$0\leq k$$, $$s\leq 1$$ and $$p, q> 1$$.
(iii) $$f(x, v)= a(|x|) v^p$$, $$g(x, u)= b(|x|) u^q$$, where $$p, q> 1$$ and the functions $$a, b\in C^1([0, \infty))$$ satisfy $$a(r)$$, $$b(r)> 0$$, $$(a(r) r^\gamma)'$$, $$(b(r) r^\gamma)'\geq 0$$ for $$r> 0$$, and $$\lim_{r\to \infty} a(r) r^\gamma= \lim_{r\to \infty} b(r) r^\gamma= \infty$$ with $$\gamma= (2(p+ 1)(q+ 1)- N(pq- 1))/(p+ q+ 2)$$.
Assume that the constants $$k$$, $$p$$, $$q$$, $$s$$ satisfy ${N- 2\over 2}\leq \max\{ {1\over k- 1},\;{p+ 1\over pq- 1},\;{1\over s- 1},\;{q+ 1\over pq- 1}\}$ in the case (i) and ${N\over 2}< \min \{{(p+ 1) (q+ 1)+ k(q+ 1)- sk\over pq- (1- s) (1- k)}, {(p+ 1) (q+ 1)+ k(q+ 1)+ s(p+ 1)\over pq+ k(q+ 1)+ s(p+ 1)- 1}\}$ in the case (ii). Then it is proved that the elliptic system has no positive classical solutions in the first case and no positive radial solutions of class $$C^2(\mathbb{R}^N)$$ in the remaining cases (ii) and (iii).
In the last section, the author gives some nonexistence results for positive solutions of the above system in a smooth bounded domain with boundary data $$u= v= 0$$. Here it is not assumed that the system has Lagrangian structure.

### MSC:

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

### Keywords:

nonexistence results