A priori estimates for a semilinear elliptic system without variational structure and their applications (Q1849479)

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 \(u+v\to\infty\).