Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems

López-Gómez, Julián

doi:10.1090/S0002-9947-99-02352-1

Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems
HTML articles powered by AMS MathViewer

by Julián López-Gómez PDF

Trans. Amer. Math. Soc. 352 (2000), 1825-1858 Request permission

Abstract:

In this work we analyze the existence, multiplicity and stability of positive solutions for a class of indefinite superlinear elliptic boundary value problems. The main contribution of this paper consists in the change of mind inherent to the fact of adding the superlinear amplitude $\varepsilon$ as an unfolding parameter. This change of mind allows us to unify many previous results obtained separately in the literature, it helps us to realize the global structure of the set of positive steady states, and provides us with a great variety of new general results. Our techniques can be applied to much more general equations and systems.

References

Stanley Alama and Manuel Del Pino, Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 1, 95–115 (English, with English and French summaries). MR 1373473, DOI 10.1016/S0294-1449(16)30098-1
Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. MR 415432, DOI 10.1137/1018114
Henri Berestycki, Italo Capuzzo-Dolcetta, and Louis Nirenberg, Problèmes elliptiques indéfinis et théorèmes de Liouville non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 10, 945–950 (French, with English and French summaries). MR 1249366
Henri Berestycki, Italo Capuzzo-Dolcetta, and Louis Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl. 2 (1995), no. 4, 553–572. MR 1356874, DOI 10.1007/BF01210623
H. Berestycki, I. Capuzzo-Dolcetta, and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 59–78. MR 1321809, DOI 10.12775/TMNA.1994.023
H. Berestycki, L. Nirenberg, and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994), no. 1, 47–92. MR 1258192, DOI 10.1002/cpa.3160470105
Haïm Brezis and Luc Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), no. 1, 55–64. MR 820658, DOI 10.1016/0362-546X(86)90011-8
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180. MR 341212, DOI 10.1007/BF00282325
E. N. Dancer, Some remarks on classical problems and fine properties of Sobolev spaces, Differential Integral Equations 9 (1996), no. 3, 437–446. MR 1371700
Manuel A. del Pino, Positive solutions of a semilinear elliptic equation on a compact manifold, Nonlinear Anal. 22 (1994), no. 11, 1423–1430. MR 1280207, DOI 10.1016/0362-546X(94)90121-X
Manuel A. del Pino and Patricio L. Felmer, Multiple solutions for a semilinear elliptic equation, Trans. Amer. Math. Soc. 347 (1995), no. 12, 4839–4853. MR 1303117, DOI 10.1090/S0002-9947-1995-1303117-5
J. Esquinas and J. López-Gómez, Multiparameter bifurcation for some particular reaction-diffusion systems, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), no. 1-2, 135–143. MR 1007540, DOI 10.1017/S0308210500028201
José M. Fraile, Pablo Koch Medina, Julián López-Gómez, and Sandro Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations 127 (1996), no. 1, 295–319. MR 1387267, DOI 10.1006/jdeq.1996.0071
Martin Golubitsky and David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. I, Applied Mathematical Sciences, vol. 51, Springer-Verlag, New York, 1985. MR 771477, DOI 10.1007/978-1-4612-5034-0
Zhong Hua Yang and H. B. Keller, A direct method for computing higher order folds, SIAM J. Sci. Statist. Comput. 7 (1986), no. 2, 351–361. MR 833908, DOI 10.1137/0907024
W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck, One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, Springer-Verlag, Berlin, 1986. MR 839450, DOI 10.1007/BFb0074922
Julián López-Gómez, On linear weighted boundary value problems, Partial differential equations (Han-sur-Lesse, 1993) Math. Res., vol. 82, Akademie-Verlag, Berlin, 1994, pp. 188–203. MR 1322747
Julián López-Gómez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations 127 (1996), no. 1, 263–294. MR 1387266, DOI 10.1006/jdeq.1996.0070
Julián López-Gómez, On the uniqueness of positive solutions for a class of sublinear elliptic problems, Boll. Un. Mat. Ital. B (7) 11 (1997), no. 3, 697–711 (English, with Italian summary). MR 1479518
Julián López-Gómez and Marcela Molina-Meyer, The maximum principle for cooperative weakly coupled elliptic systems and some applications, Differential Integral Equations 7 (1994), no. 2, 383–398. MR 1255895
W. J. Trjitzinsky, General theory of singular integral equations with real kernels, Trans. Amer. Math. Soc. 46 (1939), 202–279. MR 92, DOI 10.1090/S0002-9947-1939-0000092-6
Tiancheng Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u+hu^p=0$ on compact manifolds. II, Indiana Univ. Math. J. 40 (1991), no. 3, 1083–1141. MR 1129343, DOI 10.1512/iumj.1991.40.40049
M. Picone, Sui valori eccenzionali di un parametro da cui dipende un’equazione differenziale ordinaria del secondo ordine, Ann. Sc. Nor. Sup. Pisa 11 (1910), 1–141.
R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86
Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI 10.1016/0022-1236(71)90030-9