Opuscula Math. 40, no. 5 (2020), 537-548
https://doi.org/10.7494/OpMath.2020.40.5.537
Opuscula Mathematica
Existence results for a sublinear second order Dirichlet boundary value problem on the half-line
Dahmane Bouafia
Toufik Moussaoui
Abstract. In this paper we study the existence of nontrivial solutions for a boundary value problem on the half-line, where the nonlinear term is sublinear, by using Ekeland's variational principle and critical point theory.
Keywords: Ekeland's variational principle, critical point.
Mathematics Subject Classification: 34B40, 35A15, 35B38, 45C05, 34B24, 46T20.
- G.A. Afrouzi, A. Hadjian, V.D. Rădulescu, Variational analysis for Dirichlet impulsive differential equations with oscillatory nonlinearity, Portugal. Math. (N.S.) 70, Fasc. 3, (2013), 225-242.
- K. Ait-Mahiout, S. Djebali, T. Moussaoui, Multiple solutions for a BVP on \((0,+\infty)\) via Morse theory and \(H^{1}_{0,p}(\mathbb{R}^{+})\) versus \(C^{1}_{p}(\mathbb{R}^{+})\) local minimizers, Arab. J. Math. (2016), 5: 9-22.
- M. Badiale, E. Serra, Semilinear Elliptic Equations for Beginners Springer, New York, 2011.
- D. Bouafia, T. Moussaoui, D. O'Regan, Existence of solutions for a second order problem on the half-line via Ekeland's variational principle, Discuss. Math. Differ. Incl. Control Optim. 36 (2016), 131-140.
- M. Briki, S. Djebali, T. Moussaoui, Solvability of an Impulsive Boundary Value Problem on The Half-Line Via Critical Point Theory, Bull. Iranian Math. Soc. 43 (2017) 3, 601-615.
- S. Djebali, T. Moussaoui, A class of second order BVPs on infinite intervals, Electron. J. Qual. Theory Differ. Equ. 4 (2006), 1-19.
- S. Djebali, S. Zahar, Bounded solutios for a derivative dependent boundary value problem on the half-line, Dynam. Systems Appl. 19 (2010), 545-556.
- S. Djebali, O. Saifi, S. Zahar, Upper and lower solutions for BVPs on the half-line with variable coefficient and derivative depending nonlinearity, Electron. J. Qual. Theory Differ. Equ. (2011), no. 14, 1-18.
- S. Djebali, O. Saifi, S. Zahar, Singular boundary value problems with variable coefficients on the positive half-line, Electron. J. Differential Equations 2013 (2013), no. 73, 1-18.
- I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353.
- Y. Jabri, The Mountan Pass Theorem, Variants, Generalizations and Some Applications, Cambridge University Press, New York, 2003.
- Y. Liu, Existence and unboundedness of positive solutions for singular boundary value problems on half-line, Appl. Math. Comput. 144 (2003), 543-556.
- H. Lian, W. Ge, Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line, J. Math. Anal. Appl. 321 (2006), 781-792.
- H. Lian, W. Ge, Solvability for second-order three-point boundary value problems on a half-line, Appl. Math. Lett. 19 (2006), 1000-1006.
- H. Lian, P. Wang, W. Ge, Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals, Nonlinear Anal. 70 (2009), 2627-2633.
- R. Ma, Positive solutions for second order three-point boundary value problems, Appl. Math. Lett. 14 (2001) 1, 1-5.
- K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006), 246-255.
- Y. Tian, W. Ge, W. Shan, Positive solutions for three-point boundary value problem on the half-line, Comput. Math. Appl. 53 (2007), 1029-1039.
- B. Yan, D. O'Regan, R. Agarwal, Unbounded solutions for singular boundary value problems on the semi-infinite interval: Upper and lower solutions and multiplicity, J. Comput. Appl. Math. 197 (2006), 365-386.
- Dahmane Bouafia (corresponding author)
https://orcid.org/0000-0002-4471-9338
- University of M'sila, Department of Mathematics, M'sila, Algeria
- Toufik Moussaoui
- Laboratory of Fixed Point Theory and Applications, Department of Mathematics, E.N.S. Kouba, Algiers, Algeria
- Communicated by Binlin Zhang.
- Received: 2019-10-26.
- Revised: 2020-08-17.
- Accepted: 2020-08-19.
- Published online: 2020-10-10.