zbMATH — the first resource for mathematics

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

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
Full Text: DOI EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.