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Non-Zero Radial Solutions for Elliptic Systems with Coupled Functional BCs in Exterior Domains

Part of: Elliptic equations and systems Nonlinear integral equations Boundary value problems

Published online by Cambridge University Press:  30 January 2019

Filomena Cianciaruso ,
Gennaro Infante  and
Paolamaria Pietramala
Filomena Cianciaruso
Affiliation:
Dipartimento di Matematica e Informatica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy ([email protected]; [email protected]; [email protected])
Gennaro Infante
Affiliation:
Dipartimento di Matematica e Informatica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy ([email protected]; [email protected]; [email protected])
Paolamaria Pietramala
Affiliation:
Dipartimento di Matematica e Informatica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy ([email protected]; [email protected]; [email protected])
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Abstract

We prove new results on the existence, non-existence, localization and multiplicity of non-trivial radial solutions of a system of elliptic boundary value problems on exterior domains subject to non-local, nonlinear, functional boundary conditions. Our approach relies on fixed point index theory. As a by-product of our theory we provide an answer to an open question posed by do Ó, Lorca, Sánchez and Ubilla. We include some examples with explicit nonlinearities in order to illustrate our theory.

Keywords

elliptic systemfixed point indexconenon-trivial solutionnonlinear functional boundary conditions

MSC classification

Primary: 35J66: Nonlinear boundary value problems for nonlinear elliptic equations
Secondary: 45G15: Systems of nonlinear integral equations 34B10: Nonlocal and multipoint boundary value problems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 747 - 769
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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References

1.Abebe, A., Chhetri, M., Sankar, L. and Shivaji, R., Positive solutions for a class of superlinear semipositone systems on exterior domains, Bound. Value Probl. 2014(198) (2014), 9.Google Scholar
2.Aftalion, A. and Busca, J., Symétrie radiale pour des problèmes elliptiques surdéterminés posés dans des domaines extérieurs, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 633–638.Google Scholar
3.Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev. 18 (1976), 620–709.Google Scholar
4.Butler, D., Ko, E., Lee, E. K., Kyoung, E. and Shivaji, R., Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions, Commun. Pure Appl. Anal. 13 (2014), 2713–2731.Google Scholar
5.Castro, A., Sankar, L. and Shivaji, R., Uniqueness of nonnegative solutions for semipositone problems on exterior domains, J. Math. Anal. Appl. 394 (2012), 432–437.Google Scholar
6.Cheng, X. and Zhong, C., Existence of positive solutions for a second-order ordinary differential system, J. Math. Anal Appl. 312 (2005), 14–23.Google Scholar
7.Cianciaruso, F. and Pietramala, P., Multiple positive solutions of a (p 1, p 2)-Laplacian system with nonlinear BCs, Bound. Value Probl. 2015(163) (2015), 18.Google Scholar
8.Cianciaruso, F., Infante, G. and Pietramala, P., Solutions of perturbed Hammerstein integral equations with applications, Nonlinear Anal. Real World Appl. 33 (2017), 317–347.Google Scholar
9.Dhanya, R., Morris, Q. and Shivaji, R., Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball, J. Math. Anal. Appl. 434 (2016), 1533–1548.Google Scholar
10.Djebali, S. and Orpel, A., The continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains, Nonlinear Anal. 73 (2010), 660–672.Google Scholar
11.do Ó, J. M., Lorca, S. and Ubilla, P., Local superlinearity for elliptic systems involving parameters, J. Differ. Equ. 211 (2005), 1–19.Google Scholar
12.do Ó, J. M., Lorca, S., Sánchez, J. and Ubilla, P., Positive radial solutions for some quasilinear elliptic systems in exterior domains, Commun. Pure Appl. Anal. 5 (2006), 571–581.Google Scholar
13.do Ó, J. M., Lorca, S., Sánchez, J. and Ubilla, P., Non-homogeneous elliptic equations in exterior domains, Proc. Roy. Soc. Edinb. Sect. A 136 (2006), 139–147.Google Scholar
14.do Ó, J. M., Lorca, S., Sánchez, J. and Ubilla, P., Positive solutions for a class of multiparameter ordinary elliptic systems, J. Math. Anal. Appl. 332 (2007), 1249–1266.Google Scholar
15.do Ó, J. M., Lorca, S., Sánchez, J. and Ubilla, P., Superlinear ordinary elliptic systems involving parameters, Mat. Contemp. 32 (2007), 107–127.Google Scholar
16.do Ó, J. M., Lorca, S., Sánchez, J. and Ubilla, P., Positive solutions for some nonlocal and nonvariational elliptic systems, Complex Var. Elliptic Equ. 61 (2016), 297–314.Google Scholar
17.Franco, D., Infante, G. and O'Regan, D., Nontrivial solutions in abstract cones for Hammerstein integral systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 14 (2007), 837–850.Google Scholar
18.Gidas, B., Ni, W.-M. and Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.Google Scholar
19.Goodrich, C. S., Nonlocal systems of BVPs with asymptotically superlinear boundary conditions, Comment. Math. Univ. Carolin. 53 (2012), 79–97.Google Scholar
20.Goodrich, C. S., Nonlocal systems of BVPs with asymptotically sublinear boundary conditions, Appl. Anal. Discrete Math. 6 (2012), 174–193.Google Scholar
21.Guo, D. and Lakshmikantham, V., Nonlinear problems in abstract cones (Academic Press, 1988).Google Scholar
22.Han, G. and Wang, J., Multiple positive radial solutions of elliptic equations in an exterior domain, Monatsh. Math. 148 (2006), 217–228.Google Scholar
23.Infante, G. and Pietramala, P., Eigenvalues and non-negative solutions of a system with nonlocal BCs, Nonlinear Stud. 16 (2009), 187–196.Google Scholar
24.Infante, G. and Pietramala, P., Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations, Nonlinear Anal. 71 (2009), 1301–1310.Google Scholar
25.Infante, G. and Pietramala, P., Multiple nonnegative solutions of systems with coupled nonlinear boundary conditions, Math. Methods Appl. Sci. 37 (2014), 2080–2090.Google Scholar
26.Infante, G. and Pietramala, P., Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains, NoDEA Nonlinear Differ. Equ. Appl. 22 (2015), 979–1003.Google Scholar
27.Infante, G. and Webb, J. R. L., Nonzero solutions of Hammerstein integral equations with discontinuous kernels, J. Math. Anal. Appl. 272 (2002), 30–42.Google Scholar
28.Infante, G. and Webb, J. R. L., Three point boundary value problems with solutions that change sign, J. Integral Equations Appl. 15 (2003), 37–57.Google Scholar
29.Infante, G., Minhós, F. M. and Pietramala, P., Non-negative solutions of systems of ODEs with coupled boundary conditions, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4952–4960.Google Scholar
30.Infante, G., Pietramala, P. and Tojo, F. A. F., Nontrivial solutions of local and nonlocal Neumann boundary value problems, Proc. Roy. Soc. Edinb. Sect. A 146 (2016), 337–369.Google Scholar
31.Kang, P. and Wei, Z., Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations, Nonlinear Anal. 70 (2009), 444–451.Google Scholar
32.Karakostas, G. L., Existence of solutions for an n-dimensional operator equation and applications to BVPs, Electron. J. Differ. Equ. 17 (2014), 1–17.Google Scholar
33.Ko, E., Ramaswamy, M. and Shivaji, R., Uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball, J. Math. Anal. Appl. 423 (2015), 399–409.Google Scholar
34.Krasnosel'skiĭ, M. A. and Zabreĭko, P. P., Geometrical methods of nonlinear analysis (Springer-Verlag, Berlin, 1984).Google Scholar
35.Lan, K. Q., Multiple positive solutions of Hammerstein integral equations with singularities, Diff. Eqns and Dynam. Syst. 8 (2000), 175–195.Google Scholar
36.Lan, K. Q., Multiple positive solutions of semilinear differential equations with singularities, J. Lond. Math. Soc. 63 (2001), 690–704.Google Scholar
37.Lan, K. Q. and Lin, W., Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli, Nonlinear Anal. 74 (2011), 7184–7197.Google Scholar
38.Lan, K. Q. and Webb, J. R. L., Positive solutions of semilinear differential equations with singularities, J. Differ. Equ. 148 (1998), 407–421.Google Scholar
39.Lee, Y. H., A multiplicity result of positive radial solutions for a multiparameter elliptic system on an exterior domain, Nonlinear Anal. 45 (2001), 597–611.Google Scholar
40.Lee, E. K., Shivaji, R. and Son, B., Positive radial solutions to classes of singular problems on the exterior domain of a ball, J. Math. Anal. Appl. 434 (2016), 1597–1611.Google Scholar
41.Orpel, A., Nonlinear BVPS with functional parameters, J. Differ. Equ. 246 (2009), 1500–1522.Google Scholar
42.Orpel, A., Increasing sequences of positive evanescent solutions of nonlinear elliptic equations, J. Differ. Equ. 259 (2015), 1743–1756.Google Scholar
43.Precup, R., Componentwise compression-expansion conditions for systems of nonlinear operator equations and applications, Mathematical models in engineering, biology and medicine, in: AIP Conference Proceedings, Volume 1124 (American Institute of Physics, 2009),pp. 284–293.Google Scholar
44.Precup, R., Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems, J. Math. Anal. Appl. 352 (2009), 48–56.Google Scholar
45.Sankar, L., Sasi, S. and Shivaji, R., Semipositone problems with falling zeros on exterior domains, J. Math. Anal. Appl. 401 (2013), 146–153.Google Scholar
46.Stanczy, R., Positive solutions for superlinear elliptic equations, J. Math. Anal. Appl. 283 (2003), 159–166.Google Scholar
47.Webb, J. R. L., Positive solutions of some three point boundary value problems via fixed point index theory, Nonlinear Anal. 47 (2001), 4319–4332.Google Scholar
48.Webb, J. R. L. and Infante, G., Positive solutions of nonlocal boundary value problems: a unified approach, J. Lond. Math. Soc. 74 (2006), 673–693.Google Scholar
49.Yang, Z., Positive solutions to a system of second-order nonlocal boundary value problems, Nonlinear Anal. 62 (2005), 1251–1265.Google Scholar