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A shooting argument with oscillation for semilinear elliptic radially symmetric equations

Published online by Cambridge University Press:  14 November 2011

C. K. R. T. Jones ,
T. Küpper  and
H. Plakties
C. K. R. T. Jones
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742, U.S.A.
T. Küpper
Affiliation:
Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, D 3000 Hannover, Germany
H. Plakties
Affiliation:
Universität Dortmund, FB Mathematik, D 4600 Dortmund 50, Germany
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Synopsis

A new method is developed for finding radially symmetric solutions of semilinear elliptic problems by phase space methods. The basic idea is to formulate a shooting argument from initial conditions at r = 0 which involves encoding the oscillation information about the trajectories under consideration. The main theorem is applied to a particular nonlinearity and produces a cascade of solutions wherein the multiplicity increases with the number of zeros.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 1-2 , 1988 , pp. 165 - 180
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Berestycki, H. and Lions, P. L.. Existence d'ondes solitaries des problèmes nonlinearies du type Klein-Gordon. C.R. Acad. Sci. Paris Ser. 1 Math. 287 (1972), 503–506.Google Scholar
2Berestycki, H. and Lions, P. L.. Nonlinear scalar field equations I, II. Arch. Rational Mech. Anal. 82 (1983), 347–376.CrossRefGoogle Scholar
3Berestycki, H., Lions, P. L. and Peletier, L. A.. An ODE approach to the existence of positive solutions for semilinear problems in RN. Indiana Univ. Math. J. 30 (1981), 141–157.CrossRefGoogle Scholar
4Coffman, C. V.. Uniqueness for the ground state solution for Δu – u3 = 0 and variational characterisation of other solutions. Arch. Rational Mech. Anal. 46 (1972), 21–95.CrossRefGoogle Scholar
5Conley, C.. The behaviour of spherically symmetric equilibria near infinity (MRC technical summary report 2117, 1980).Google Scholar
6Fisher, R. A.. The wave of advance of advantageous genes. J. Eugenics 7 (1937), 355–369.CrossRefGoogle Scholar
7Heinz, H. P.. Nodal properties and variational characterisations of solutions to nonlinear Sturm-Liouville problems. J. Differential Equations 62 (1986), 299–333.CrossRefGoogle Scholar
8Heinz, H. P.. Nodal properties and bifurcation from the essential spectrum for a class of nonlinear Sturm Liouville Problems. J. Differential Equations 64 (1986), 79–108.CrossRefGoogle Scholar
9Heinz, H. P.. Free Ljusternik-Schnirelman Theory and the Bifurcation Diagrams of Certain Singular Nonlinear Problems. J. Differential Equations 66 (1987), 263–300.CrossRefGoogle Scholar
10Jones, C. K. R. T.. On the infinitely many standing waves of some nonlinear schroedinger equations. Lectures in Appl. Math. 23 (1986), 3–21.Google Scholar
11Jones, C. K. R. T. and Kiipper, T.. On the infinitely many solutions of semilinear elliptic equation. SIAM J. Math. Anal. 17 (1986), 803–835.CrossRefGoogle Scholar
12Kerékjártó, B.. Vorlesungen iiber Topologie I (Berlin: Springer, 1923).CrossRefGoogle Scholar
13Lange, H.. Stationary and solitary wave type solutions of singular nonlinear schroedinger equations (preprint).Google Scholar
14Nehari, Z.. On a nonlinear differential equation arising in nuclear physics. Proc. Royal Irish Acad. Sect. A 62 (1963), 117–135.Google Scholar
15Peletier, L. A. and Serrin, J.. Uniqueness of positive solutions of semilinear equation in ℛn. Arch Rational Mech. Anal. 81 (1983), 181–197.CrossRefGoogle Scholar
16Plakties, H.. Charakterisierung der oszillierenden Losung einer nicht-Unearen Differentialgleichung (Diplomarbeit, Dortmund, 1986).Google Scholar
17Pohozaev, S. I.. Eigenfunctions of the equation Δu + λf(u) = 0. Soviet Math. Dokl. 6 (1965), 1408–1411.Google Scholar
18Ryder, G. H.. Boundary value problems for a class of nonlinear differential equations. Pacific J. Math. 22 (1967), 477–503.CrossRefGoogle Scholar
19Serrin, J.. Phase transitions and interfacial layers for van der Waals fluids: In Recent methods in nonlinear Analysis and Applications, Proceedings of the fourth international meeting of SAFA, eds. Confera, A. et al. pp. 169–175 (Neapel, 1980).Google Scholar
20Strauss, W. A.. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), 149–162.CrossRefGoogle Scholar