Semilinear elliptic equations and fixed points
HTML articles powered by AMS MathViewer
- by Cleon S. Barroso PDF
- Proc. Amer. Math. Soc. 133 (2005), 745-749 Request permission
Abstract:
In this paper, we deal with a class of semilinear elliptic equations in a bounded domain $\Omega \subset \mathbb {R}^N$, $N\geq 3$, with $C^{1,1}$ boundary. Using a new fixed point result of the Krasnoselskii type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical.References
- Michael Struwe, Variational methods, Springer-Verlag, Berlin, 1990. Applications to nonlinear partial differential equations and Hamiltonian systems. MR 1078018, DOI 10.1007/978-3-662-02624-3
- Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 845785, DOI 10.1090/cbms/065
- Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007, DOI 10.1007/978-1-4612-4146-1
- M. M. Vainberg, Variational methods for the study of nonlinear operators, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. With a chapter on Newton’s method by L. V. Kantorovich and G. P. Akilov. Translated and supplemented by Amiel Feinstein. MR 0176364
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- Cleon S. Barroso, Krasnoselskii’s fixed point theorem for weakly continuous maps, Nonlinear Anal. 55 (2003), no. 1-2, 25–31. MR 2001629, DOI 10.1016/S0362-546X(03)00208-6
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Abbas Bahri, Topological results on a certain class of functionals and application, J. Functional Analysis 41 (1981), no. 3, 397–427. MR 619960, DOI 10.1016/0022-1236(81)90083-5
- Abbas Bahri and Henri Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), no. 1, 1–32. MR 621969, DOI 10.1090/S0002-9947-1981-0621969-9
- Zhiren Jin, Multiple solutions for a class of semilinear elliptic equations, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3659–3667. MR 1443158, DOI 10.1090/S0002-9939-97-04199-3
Additional Information
- Cleon S. Barroso
- Affiliation: Departamento de Matematica, Universidade Federal do Ceará, Campus do Pici, Bl. 914, Fortaleza-Ce, 60455-760, Brazil
- Email: cleonbar@mat.ufc.br
- Received by editor(s): September 25, 2003
- Published electronically: October 21, 2004
- Additional Notes: This research was supported by Capes, Brazil
- Communicated by: David S. Tartakoff
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 745-749
- MSC (2000): Primary 35J25; Secondary 47H10
- DOI: https://doi.org/10.1090/S0002-9939-04-07718-4
- MathSciNet review: 2113923