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A priori estimates for a semilinear elliptic system without variational structure and their applications. (English) Zbl 1005.35024
The paper investigates nonnegative solutions to the semilinear elliptic system $$\Delta u+f(x,u,v)=0$$, $$\Delta v+g(x,u,v)=0$$ in a domain $$\Omega\subset \mathbb{R}^N$$ together with the homogeneous Dirichlet boundary conditions $$u=v=0$$ on $$\partial\Omega$$. First, the author proves an $$L^\infty$$ estimate for a solution to the above system whenever $$N=3$$, $$f(x,u,v)=au^r+bv^q$$, $$g(x,u,v)=cu^p+dv^s$$ provided (i) $$\alpha+\beta>1$$, (ii) $$\max[r,s]<5$$, (iii) $$\beta\not= 2/(r-1)$$ and $$\alpha\not= 2/(s-1)$$, where $$a$$, $$b$$, $$c$$, $$d$$ and $$p$$, $$q$$, $$r$$, $$s$$ are non negative numbers satisfying $$pq>1$$, $$r,s>1$$, and $$\alpha=2(p+1)/(pq-1)$$, $$\beta=2(q+1)/(pq-1)$$. Then, by using the above a priori estimate and a fixed point theorem, he proves an existence result under the previous assumptions and the extra conditions $$p,q>1$$, $$a+b>0$$, $$c+d>0$$. Furthermore, the author discusses the general case $$N\geq 3$$ with more general nonlinearities $$f(x,u,v)$$ and $$g(x,u,v)$$ satisfying suitable growth conditions as $$u+v\to\infty$$.
Reviewer: G.Porru (Cagliari)

##### MSC:
 35B45 A priori estimates in context of PDEs 35J55 Systems of elliptic equations, boundary value problems (MSC2000)
##### Keywords:
elliptic systems; a priori estimates
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