Nonzero positive solutions of a multi-parameter elliptic system with functional BCs

Gennaro Infante

DOI: http://dx.doi.org/10.12775/TMNA.2017.060

Abstract


We prove, by topological methods, new results on the existence of nonzero positive weak solutions for a class of multi-parameter second order elliptic systems subject to functional boundary conditions. The setting is fairly general and covers the case of multi-point, integral and nonlinear boundary conditions. We also present a non-existence result. We provide some examples to illustrate the applicability of our theoretical results.

Keywords


Positive solution; elliptic system; functional boundary condition; cone; fixed point index

Full Text:

PREVIEW FULL TEXT

References


H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal. 11 (1972), 346–384.

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM. Rev. 18 (1976), 620–709.

A. Cabada, An overview of the lower and upper solutions method with nonlinear boundary value conditions, Bound. Value Probl. (2011), Art. ID 893753, 18 pp.

F. Cianciaruso, G. Infante and P. Pietramala, Solutions of perturbed Hammerstein integral equations with applications, Nonlinear Anal. Real World Appl. 33 (2017), 317–347.

J.A. Cid and G. Infante, A non-variational approach to the existence of nonzero positive solutions for elliptic systems, J. Fixed Point Theory Appl. 19 (2017), 3151–3162.

R. Conti, Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Unione Mat. Ital. 22 (1967), 135–178.

J.D.B. de Godoi, O.H. Miyagaki and R.S. Rodrigues, A class of nonlinear elliptic systems with Steklov–Neumann nonlinear boundary conditions, Rocky Mountain J. Math. 46 (2016), 1519–1545.

D.R. Dunninger and H. Wang, Multiplicity of positive solutions for a nonlinear differential equation with nonlinear boundary conditions, Ann. Polon. Math. 69 (1998), 155–165.

C.S. Goodrich, Perturbed Hammerstein integral equations with sign-changing kernels and applications to nonlocal boundary value problems and elliptic PDEs, J. Integral Equations Appl. 28 (2016), 509–549.

C.S. Goodrich, A new coercivity condition applied to semipositone integral equations with nonpositive, unbounded nonlinearities and applications to nonlocal BVPs, J. Fixed Point Theory Appl. 19 (2017), 1905–1938.

C.S. Goodrich, The effect of a nonstandard cone on existence theorem applicability in nonlocal boundary value problems, J. Fixed Point Theory Appl. 19 (2017), 2629–2646.

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, 1988.

G.L. Karakostas and P.Ch. Tsamatos, Existence of multiple positive solutions for a nonlocal boundary value problem, Topol. Methods Nonlinear Anal. 19 (2002), 109–121.

G.L. Karakostas and P.Ch. Tsamatos, Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems, Electron. J. Differential Equations 2002, 17 pp.

K.Q. Lan, Nonzero positive solutions of systems of elliptic boundary value problems, Proc. Amer. Math. Soc. 139 (2011), 4343–4349.

K.Q. Lan, Existence of nonzero positive solutions of systems of second order elliptic boundary value problems, J. Appl. Anal. Comput. 1 (2011), 21–31.

R. Ma, A survey on nonlocal boundary value problems, Appl. Math. E-Notes 7 (2007), 257–279.

R. Ma, R. Chen and Y. Lu, Method of lower and upper solutions for elliptic systems with nonlinear boundary condition and its applications, J. Appl. Math. (2014), Art. ID 705298, 7 pp.

J. Mawhin and K. Schmitt, Upper and lower solutions and semilinear second order elliptic equations with non-linear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 199–207.

S.K. Ntouyas, Nonlocal initial and boundary value problems: a survey, Handbook of Differential Equations: Ordinary Differential Equations, Vol. II, Elsevier B.V., Amsterdam, 2005, 461–557.

C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.

C.V. Pao and Y.M. Wang, Nonlinear fourth-order elliptic equations with nonlocal boundary conditions, J. Math. Anal. Appl. 372 (2010), 351–365.

M. Picone, Su un problema al contorno nelle equazioni differenziali lineari ordinarie del secondo ordine, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (1908), 1–95.

A. Štikonas, A survey on stationary problems, Green’s functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions, Nonlinear Anal. Model. Control 19 (2014), 301–334.

J.R.L. Webb, Existence of positive solutions for a thermostat model, Nonlinear Anal. Real World Appl. 13 (2012), 923–938.

J.R.L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc. 74 (2006), 673–693.

W.M. Whyburn, Differential equations with general boundary conditions, Bull. Amer. Math. Soc. 48 (1942), 692–704.

E. Zeidler, Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems, Springer, New York, 1986.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism