Let \(\Omega\) be a bounded domain in \(R^ N\), \(N\geq 2\), with smooth boundary \(\partial \Omega\) and let \(\partial /\partial n\) denote the outward normal derivative operator. The authors derive a priori estimates for nonnegative solutions of (*): \(d\Delta u-u+h(u)=0\) in \(\Omega\), \(\partial u/\partial n=0\) on \(\partial \Omega\), where \(d>0\) and \(h: [0,\infty)\to [0,\infty)\) is continuous and satisfies \(a_ 0u^{P_ 0}\leq h(u)\leq a_ 1u\) p for sufficiently large u with positive constants \(a_ 0\) and \(a_ 1\) independent of u and the exponents \(p_ 0\) and p satisfying \(1<p_ 0<p<N/(N-2)\). These estimates are used to prove the nonexistence of positive nonconstant solutions for sufficiently large d. The existence of nonconstant solutions near a constant solution via bifurcation theory are also deduced.
A similar analysis is performed and results obtained for systems which include \((**): d\Delta u-u+u\quad p/v\quad q+\sigma =0,\) \(D\Delta v-\nu v+u\quad r/v\quad s=0\) in \(\Omega\), \(\partial u/\partial n=0\), \(\partial v/\partial n=0\), on \(\partial \Omega\), where d, D, and \(\nu\) are positive constants, \(\sigma\) is a nonnegative constant, and the exponents satisfy \(p>1\), \(q>0\), \(r>0\), \(s\geq 0\), and \(0<(p-1)/q<r/(s+1)\). The system (**) arises in biological pattern formation theory.