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On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type. (English) Zbl 0635.35031
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.
Reviewer: P.W.Schaefer

MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35J25 Boundary value problems for second-order elliptic equations 35B32 Bifurcations in context of PDEs 92B05 General biology and biomathematics
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References:
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