Uniqueness of the solution of a semilinear boundary value problem. (English) Zbl 0576.35044
Der Autor zeigt die Eindeutigkeit der positiven Lösung des Randwertproblems
\[
\epsilon^ 2\Delta u+f(x,u)=0,\quad x\in \Omega,\quad u(x)=0,\quad x\in \partial \Omega
\]
für \(0<\epsilon \ll 1\), wobei die Nichtlinearität f die Bedingungen (i) f(x,0)\(\geq 0\), \(x\in {\bar \Omega}\); (ii) \(f_ u(x,0)>0\), wenn \(f(x,0)=0\); (iii) Es gibt \(a\in C^ 1({\bar \Omega})\), so daß \(f(x,u)>0\) für \(0<u<a(x)\) und \(f(x,u)<0\) für \(u>a(x)\); (iv) \(f_ u(x,a(x))<0\), \(x\in {\bar \Omega}\) erfüllt und wobei f und der Rand des Gebietes \(\Omega \subset {\mathbb{R}}^ n\) genügend glatt sind.
Reviewer: W.Wendt
MSC:
35J65 | Nonlinear boundary value problems for linear elliptic equations |
35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
35B50 | Maximum principles in context of PDEs |
35J25 | Boundary value problems for second-order elliptic equations |
35B40 | Asymptotic behavior of solutions to PDEs |
References:
[1] | Angenent, S.B., Mallet-Paret, J., Peletier, L.A.: Stable transitionlayers in a semilinear boundaryvalue problem. Math. Inst. Univ. Leiden Rep.7 (1984) · Zbl 0634.35041 |
[2] | Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys.68, 209-243 (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125 |
[3] | Protter, M.H., Weinberger, H.F.: Maximum principles in differential equations. Englewood Cliffs, NJ: Prentice-Hall 1967 · Zbl 0153.13602 |
[4] | Sattinger, D.H.: Topics in stability and bifurcation theory. Lect. Notes Math. 309, p. 39. Berlin, Heidelberg, New York: Springer 1973 · Zbl 0248.35003 |
[5] | Serrin, J.: A symmetry problem in potential theory. Arch. Rat. Mech. Anal.43, 304-318 (1971) · Zbl 0222.31007 · doi:10.1007/BF00250468 |
[6] | Serrin, J.: Nonlinear elliptic equations of second order. AMS Symposium in Partial Differential Equations, Berkeley, August 1971 · Zbl 0271.35004 |
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