Positive solutions of a non-linear boundary value problem
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Abstract
This paper deals with a non-linear second order ordinary dierential equation with symmetric non-linear boundary conditions, where both of the nonlinearities are of power type. It provides results concerning the existence and multiplicity of positive solutions, both symmetric and non-symmetric, for values of parameters not considered before. The main tool is the shooting method.
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How to Cite
Peres, S.
(2016).
Positive solutions of a non-linear boundary value problem.
Acta Mathematica Universitatis Comenianae, 85(2), 285-310.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/247/391
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References
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2. Arrieta J. M. and Rodriguez-Bernal A., Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions, Communications in Partial Differential Equations 29(7{8) (2004), 1127-1148.
3. Brighi B. and Ramaswamy M., On some general semilinear elliptic problems with nonlinear boundary conditions, Advances in Differential Equations 4(3) (1999), 369-390.
4. Cano-Casanova S., On the positive solutions of the logistic weighted elliptic BVP with sublinear mixed boundary conditions, Cano-Casanova, S. Lopez-Gomez J. and Mora-Corral C. (eds.) Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, 1-15, World Scientific, Singapore 2005.
5. Chipot M., Fila M. and Quittner P., Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Mathematica Universitatis Comenianae 60(1) (1991), 35-103.
6. Chipot M. and Quittner P., Equilibria, connecting orbits and a priori bounds for semilinear parabolic equations with nonlinear boundary conditions, Journal of Dynamics and Differential Equations 16(1) (2004), 91-138.
7. Chipot M. and Ramaswamy M., Semilinear elliptic problems with nonlinear boundary conditions, Differential Equations and Dynamical Systems 6(1{2) (1998), 51-75.
8. Fila M. and Kawohl B., Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition, Advances in Mathematical Sciences and Applications 11 (2001), 113-126.
9. Fila M., Velazquez J. J. L. and Winkler M., Grow-up on the boundary for a semilinear parabolic problem, Progress in Nonlinear Differential Equations and Their Applications, vol.
64: Nonlinear Elliptic and Parabolic Problems, 137-150, Birkhauser, Basel 2005.
10. Filo J., Diffusivity versus absorption through the boundary, Journal of Differential Equations 99(2) (1992), 281-305.
11. Garca-Melian J., Morales-Rodrigo C., Rossi J. D. and Suarez A., Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming
ux on the boundary, Annali di Matematica Pura ed Applicata 187(3) (2008), 459-486.
12. Johnson W. P., The Curious History of Faa di Bruno's Formula, The American Mathematical Monthly 109(3) (2002), 217-234.
13. Leung A. W. and Zhang Q., Reaction diffusion equations with nonlinear boundary conditions, blowup and steady states, Mathematical Methods in the Applied Sciences 21(17) (1998), 1593-1617.
14. Lopez Gomez J., Marquez V. and Wolanski N., Dynamic behavior of positive solutions to reaction-diffusion problems with nonlinear absorption through the boundary, Revista de la Union Matematica Argentina 38(3{4) (1993), 196-209.
15. Morales-Rodrigo C. and Suarez A., Some elliptic problems with nonlinear boundary conditions, Cano-Casanova S., Lopez-Gomez J. and Mora-Corral C. (eds.) Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, 175-199, World Scientiffc, Singapore 2005.
16. Peres S., Solvability of a nonlinear boundary value problem, Acta Mathematica Universitatis Comenianae 82(1) (2013), 69-103.
17. Peres S., Positive non-symmetric solutions of a non-linear boundary value problem, Electronic Journal of Qualitative Theory of Differential Equations, (69) (2013), 1-24.
18. Peres S., Nonsymmetric solutions of a nonlinear boundary value problem, Czechoslovak Mathematical Journal, 64(2) (2014), 495-508.
19. Quittner P., On global existence and stationary solutions for two classes of semilinear parabolic problems, Commentationes Mathematicae Universitatis Carolinae 34(1) (1993), 105-124.
20. Rodriguez-Bernal A. and Tajdine A., Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up, Journal of Differential Equations 169(2) (2001), 332-372.
21. Rossi J. D., Elliptic problems with nonlinear boundary conditions and the Sobolev trace theorem, Chipot M. and Quittner P. (eds.) Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 2, Chapter 5, 311-406, Elsevier, Amsterdam 2005.
2. Arrieta J. M. and Rodriguez-Bernal A., Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions, Communications in Partial Differential Equations 29(7{8) (2004), 1127-1148.
3. Brighi B. and Ramaswamy M., On some general semilinear elliptic problems with nonlinear boundary conditions, Advances in Differential Equations 4(3) (1999), 369-390.
4. Cano-Casanova S., On the positive solutions of the logistic weighted elliptic BVP with sublinear mixed boundary conditions, Cano-Casanova, S. Lopez-Gomez J. and Mora-Corral C. (eds.) Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, 1-15, World Scientific, Singapore 2005.
5. Chipot M., Fila M. and Quittner P., Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Mathematica Universitatis Comenianae 60(1) (1991), 35-103.
6. Chipot M. and Quittner P., Equilibria, connecting orbits and a priori bounds for semilinear parabolic equations with nonlinear boundary conditions, Journal of Dynamics and Differential Equations 16(1) (2004), 91-138.
7. Chipot M. and Ramaswamy M., Semilinear elliptic problems with nonlinear boundary conditions, Differential Equations and Dynamical Systems 6(1{2) (1998), 51-75.
8. Fila M. and Kawohl B., Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition, Advances in Mathematical Sciences and Applications 11 (2001), 113-126.
9. Fila M., Velazquez J. J. L. and Winkler M., Grow-up on the boundary for a semilinear parabolic problem, Progress in Nonlinear Differential Equations and Their Applications, vol.
64: Nonlinear Elliptic and Parabolic Problems, 137-150, Birkhauser, Basel 2005.
10. Filo J., Diffusivity versus absorption through the boundary, Journal of Differential Equations 99(2) (1992), 281-305.
11. Garca-Melian J., Morales-Rodrigo C., Rossi J. D. and Suarez A., Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming
ux on the boundary, Annali di Matematica Pura ed Applicata 187(3) (2008), 459-486.
12. Johnson W. P., The Curious History of Faa di Bruno's Formula, The American Mathematical Monthly 109(3) (2002), 217-234.
13. Leung A. W. and Zhang Q., Reaction diffusion equations with nonlinear boundary conditions, blowup and steady states, Mathematical Methods in the Applied Sciences 21(17) (1998), 1593-1617.
14. Lopez Gomez J., Marquez V. and Wolanski N., Dynamic behavior of positive solutions to reaction-diffusion problems with nonlinear absorption through the boundary, Revista de la Union Matematica Argentina 38(3{4) (1993), 196-209.
15. Morales-Rodrigo C. and Suarez A., Some elliptic problems with nonlinear boundary conditions, Cano-Casanova S., Lopez-Gomez J. and Mora-Corral C. (eds.) Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, 175-199, World Scientiffc, Singapore 2005.
16. Peres S., Solvability of a nonlinear boundary value problem, Acta Mathematica Universitatis Comenianae 82(1) (2013), 69-103.
17. Peres S., Positive non-symmetric solutions of a non-linear boundary value problem, Electronic Journal of Qualitative Theory of Differential Equations, (69) (2013), 1-24.
18. Peres S., Nonsymmetric solutions of a nonlinear boundary value problem, Czechoslovak Mathematical Journal, 64(2) (2014), 495-508.
19. Quittner P., On global existence and stationary solutions for two classes of semilinear parabolic problems, Commentationes Mathematicae Universitatis Carolinae 34(1) (1993), 105-124.
20. Rodriguez-Bernal A. and Tajdine A., Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up, Journal of Differential Equations 169(2) (2001), 332-372.
21. Rossi J. D., Elliptic problems with nonlinear boundary conditions and the Sobolev trace theorem, Chipot M. and Quittner P. (eds.) Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 2, Chapter 5, 311-406, Elsevier, Amsterdam 2005.