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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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[1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623 – 727. · Zbl 0093.10401
[2] Reinhold Böhme, Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwertprobleme, Math. Z. 127 (1972), 105 – 126 (German). · Zbl 0254.47082
[3] Haïm Brézis and Walter A. Strauss, Semi-linear second-order elliptic equations in \?\textonesuperior , J. Math. Soc. Japan 25 (1973), 565 – 590. · Zbl 0278.35041
[4] Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321 – 340. · Zbl 0219.46015
[5] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972), 30-39. · Zbl 0297.92007
[6] C. E. Kenig, Private communications.
[7] J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Math. 8 (1939), 78-91. · JFM 65.0257.02
[8] H. Meinhardt, Models of biological pattern formation, Academic Press, London and New York, 1982.
[9] Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0142.38701
[10] Wei Ming Ni, On the positive radial solutions of some semilinear elliptic equations on \?\(^{n}\), Appl. Math. Optim. 9 (1983), no. 4, 373 – 380. · Zbl 0527.35026
[11] Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal. 13 (1982), no. 4, 555 – 593. · Zbl 0501.35010
[12] Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487 – 513. · Zbl 0212.16504
[13] Paul H. Rabinowitz, A bifurcation theorem for potential operators, J. Functional Analysis 25 (1977), no. 4, 412 – 424. · Zbl 0369.47038
[14] Franz Rothe, Global solutions of reaction-diffusion systems, Lecture Notes in Mathematics, vol. 1072, Springer-Verlag, Berlin, 1984. · Zbl 0546.35003
[15] G. Stampacchia, Équations elliptiques à données discontinues, Séminaire Schwartz, 1960/61, no. 4. · Zbl 0092.10401
[16] Izumi Takagi, A priori estimates for stationary solutions of an activator-inhibitor model due to Gierer and Meinhardt, Tôhoku Math. J. (2) 34 (1982), no. 1, 113 – 132. · Zbl 0493.35004
[17] Izumi Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations 61 (1986), no. 2, 208 – 249. · Zbl 0627.35049
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