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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 $\Omega\subset \bbfR^N$ 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 $N\ge 3$ with more general nonlinearities $f(x,u,v)$ and $g(x,u,v)$ satisfying suitable growth conditions as $u+v\to\infty$.

35B45A priori estimates for solutions of PDE
35J55Systems of elliptic equations, boundary value problems (MSC2000)
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