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Existence and non-existence results for semilinear elliptic problems in unbounded domains

Published online by Cambridge University Press:  14 November 2011

M. J. Esteban  and
P. L. Lions
M. J. Esteban
Affiliation:
Laboratoire d'Analyse Numérique, Université P. et M. Curie, 4 Place Jussieu, 75230 Paris Cedex, France
P. L. Lions
Affiliation:
C.N.R.S. Laboratoire d'Analyse Numérique, Université P. et M. Curie, 4 Place Jussieu, 75230 Paris Cedex, France
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Synopsis

In this paper, we prove various existence and non-existence results for semilinear elliptic problems in unbounded domains. In particular we prove for general classes of unbounded domains that there exists no solution distinct from 0 of

for any smooth f satisfying f(0) = 0. This result is obtained by the use of new identities that solutions of semilinear elliptic equations satisfy.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 93 , Issue 1-2 , 1982 , pp. 1 - 14
Copyright
Copyright © Royal Society of Edinburgh 1982

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References

1Amann, H.. Nonlinear operators in ordered Banach spaces and some applications to nonlinear value problems. In Nonlinear operators and the calculus of variations (Bruxelles). Lecture Notes in Mathematics 53, 1–55 (New York: Springer, 1975).Google Scholar
2Ambrosetti, A. and Rabinowitz, P. H.. Dual variational methods in critical points theory and applications. J. Funct. Anal. 14 (1973), 349–381.CrossRefGoogle Scholar
3Berestycki, H. and Lions, P. L.. Existence of solutions for nonlinear scalar field equations. Part. I: The ground state. Part. II: Existence of infinitely many bound states. Arch. Rational Mech. Anal., to appear.Google Scholar
4Berestycki, H. and Lions, P. L.. Existence d'ondes solitaires dans des problèmes nonlinéaires du type Klein-Gordon. C. R. Acad. Set. Paris 287 (1978), 503–506; 288 (1979), 395–398.Google Scholar
5Berestycki, H. and Lions, P. L.. To appear in Comm. Math. Phys.Google Scholar
6Berestycki, H. and Lions, P. L.. Existence of stationary states of nonlinear scalar field equations. In Bifurcation phenomena in mathematical physics and related topics, ed. Bardos, C. and Bessis, D. (New York: Reidel, 1980).Google Scholar
7Berestycki, H. and Lions, P. L.. Existence of a ground state in nonlinear equations of the type Klein-Gordon. In Variational Inequalities, ed. Cottle, , Gianessi, and Lions, (New York: Wiley, 1979).Google Scholar
8Berestycki, H. and Lions, P. L.. Une méthode locale pour l'existence de solutions positives de problèmes seminlinéaires elliptiques dans ℝN. J. Analyse Math. 38 (1980), 144–187.CrossRefGoogle Scholar
9Berestycki, 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
10Berger, M. S.. On the existence and structure of stationary states for a nonlinear Klein–Gordon equation. J. Funct. Anal. 9 (1972), 249–261.CrossRefGoogle Scholar
11Bona, J. and Turner, R. E. L.. Personnal communication.Google Scholar
12Brezis, H. and Turner, R. E. L.. On a class of superlinear elliptic problems. Comm. Partial Differential Equations 2 (1977), 601–614.CrossRefGoogle Scholar
13Coffman, C. V.. Uniqueness of the ground state solution for Δu − u + u 3 = 0 and a variational characterization of other solutions. Arch. Rational Mech. Anal. 46 (1972), 81–95.CrossRefGoogle Scholar
14Coleman, S., Glazer, V. and Martin, A.. Action minima among solutions to a class of Euclidean scalar field equations. Comm. Math. Phys. 58 (1978), 211–221.CrossRefGoogle Scholar
15Esteban, M. J.. Existence d'une infinité d'ondes solitaires pour des equations de Champs non linéaires. Ann. Fac. Sci. Toulouse Math. 2 (1980), 3–4.CrossRefGoogle Scholar
16Esteban, M. J.. Nonlinear elliptic problems in strip-like domains, to appear, see also thèse de 3 ème cycle, Paris VI (1981).Google Scholar
17De Figueiredo, D. G., Lions, P. L. and Nussbaum, R. D.. A priori estimates for positive solutions of semilinear elliptic equations. J. Math. Pures Appl., (1981), to appear.CrossRefGoogle Scholar
18Gidas, B.. Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations. In Nonlinear Partial Differential Equations in Engineering and Applied Science, ed. Sternberg, R., Kalinowski, A. and Papadakis, J. (New York: Dekker, 1980).Google Scholar
19Gidas, B., Ni, W. and Nirenberg, L.. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
20Gidas, B., Ni, W. and Nirenberg, L.. Adv. in Math. (1981), to appear.Google Scholar
21Lions, P. L.. On positive solutions of semilinear elliptic equations, to appear; see also MRC report, Univ. of Wisconsin-Madison, (1981).CrossRefGoogle Scholar
22Pohozaev, S. I.. Eigenfunctions of the equation Δu + λf(u) = 0. Soviet Math. Dokl. 5 (1965), 1408–1411.Google Scholar
23Rabinowitz, P. H.. Variational methods for nonlinear eigenvalue problems. In Eigenvalues of nonlinear problems, C.I.M.E. (Rome: Edizioni Cremonese, 1974).Google Scholar
24Strauss, W.. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), 154–162.CrossRefGoogle Scholar