Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
nonlinear second order ordinary differential equation; existence of solution; multiplicity of solution; nonlinear boundary condition; shooting method; time map
nonlinear second order ordinary differential equation; existence of solution; multiplicity of solution; nonlinear boundary condition; shooting method; time map
Summary:
We study the existence and multiplicity of positive nonsymmetric and sign-changing nonantisymmetric solutions of a nonlinear second order ordinary differential equation with symmetric nonlinear boundary conditions, where both of the nonlinearities are of power type. The given problem has already been studied by other authors, but the number of its positive nonsymmetric and sign-changing nonantisymmetric solutions has been determined only under some technical conditions. It was a long-standing open question whether or not these conditions can be omitted. In this article we provide the answer. Our main tool is the shooting method.
References:
[1] Andreu, F., Mazón, J. M., Toledo, J., Rossi, J. D.: Porous medium equation with absorption and a nonlinear boundary condition. Nonlinear Anal., Theory Methods Appl., Ser. A 49 (2002), 541-563. DOI 10.1016/S0362-546X(01)00122-5 | MR 1888335 | Zbl 1011.35109
[2] Arrieta, J. M., Rodríguez-Bernal, A.: Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions. Commun. Partial Differ. Equations 29 (2004), 1127-1148. DOI 10.1081/PDE-200033760 | MR 2097578 | Zbl 1058.35119
[3] Brighi, B., Ramaswamy, M.: On some general semilinear elliptic problems with nonlinear boundary conditions. Adv. Differ. Equ. 4 (1999), 369-390. MR 1671255 | Zbl 0948.35047
[4] Cano-Casanova, S.: On the positive solutions of the logistic weighted elliptic BVP with sublinear mixed boundary conditions. Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology. Proceedings of the international seminar on spectral theory and nonlinear analysis, Universidad Complutense de Madrid, 2004 S. Cano-Casanova et al. World Scientific, Hackensack (2005), 1-15. MR 2328349 | Zbl 1137.35373
[5] Chipot, M., Fila, M., Quittner, P.: Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. Acta Math. Univ. Comen., New Ser. 60 (1991), 35-103. MR 1120596 | Zbl 0743.35038
[6] Chipot, M., Quittner, P.: Equilibria, connecting orbits and a priori bounds for semilinear parabolic equations with nonlinear boundary conditions. J. Dyn. Differ. Equations 16 (2004), 91-138. DOI 10.1023/B:JODY.0000041282.14930.7a | MR 2093837 | Zbl 1077.35065
[7] Chipot, M., Ramaswamy, M.: Semilinear elliptic problems with nonlinear boundary conditions. Differ. Equ. Dyn. Syst. 6 (1998), 51-75. MR 1660245 | Zbl 0985.35032
[8] Fila, M., Kawohl, B.: Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition. Adv. Math. Sci. Appl. 11 (2001), 113-126. MR 1841563 | Zbl 1194.35055
[9] Fila, M., Velázquez, J. J. L., Winkler, M.: Grow-up on the boundary for a semilinear parabolic problem. Nonlinear elliptic and parabolic problems: a special tribute to the work of Herbert Amann. Proc. Conf., Zürich, 2004 M. Chipot, J. Escher Progress in Nonlinear Differential Equations and Their Applications 64 Birkhäuser, Basel (2005), 137-150. MR 2185214 | Zbl 1090.35034
[10] García-Melián, J., Morales-Rodrigo, C., Rossi, J. D., Suárez, A.: Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary. Ann. Mat. Pura Appl. (4) 187 (2008), 459-486. MR 2393144 | Zbl 1150.35011
[11] Leung, A. W., Zhang, Q.: Reaction diffusion equations with non-linear boundary conditions, blowup and steady states. Math. Methods Appl. Sci. 21 (1998), 1593-1617. DOI 10.1002/(SICI)1099-1476(19981125)21:17<1593::AID-MMA12>3.0.CO;2-1 | MR 1654625
[12] Gómez, J. López, Márquez, V., Wolanski, N.: Dynamic behavior of positive solutions to reaction-diffusion problems with nonlinear absorption through the boundary. Rev. Unión Mat. Argent. 38 (1993), 196-209. MR 1276024
[13] Morales-Rodrigo, C., Suárez, A.: Some elliptic problems with nonlinear boundary conditions. Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology. Proceedings of the international seminar on spectral theory and nonlinear analysis, Universidad Complutense de Madrid, 2004 S. Cano-Casanova et al. World Scientific, Hackensack (2005), 175-199. MR 2328357 | Zbl 1133.35369
[14] Peres, S.: Solvability of a nonlinear boundary value problem. Acta Math. Univ. Comen., New Ser. 82 (2013), 69-103. MR 3028151
[15] Quittner, P.: On global existence and stationary solutions for two classes of semilinear parabolic problems. Commentat. Math. Univ. Carol. 34 (1993), 105-124. MR 1240209 | Zbl 0794.35089
[16] Rodríguez-Bernal, A., Tajdine, A.: Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up. J. Differ. Equations 169 (2001), 332-372. DOI 10.1006/jdeq.2000.3903 | MR 1808470 | Zbl 0999.35047
[17] Rossi, J. D.: Elliptic problems with nonlinear boundary conditions and the Sobolev trace theorem. Stationary Partial Differential Equations. Vol. II. M. Chipot, P. Quittner Handbook of Differential Equations Elsevier/North Holland, Amsterdam (2005), 311-406. MR 2181485 | Zbl 1207.35160
Search
Partner of
