Energy-based localization of positive solutions for stationary Kirchhoff-type equations and systems


In this paper, we are concerned with positive solutions for the Dirichlet boundary value problem for equations and systems of Kirchhoff type. We obtain existence and localization results of positive solutions using Krasnosel’skiĭ’s fixed point theorem in cones and a weak Harnack-type inequality. The localization is given in terms of energy norm, being of interest from a physical point of view. In the case of systems, the results on the localization are established componentwise using the vector version of Krasnosel’skiĭ’s theorem, which allows some of the equations of the system to satisfy the compression condition and others the expansion one.


Nataliia Kolun
Department of Fundamental Sciences, Military Academy, 65009 Odessa, Ukraine
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania


Kirchhoff equation; positive solution; Dirichlet boundary value problem; Krasnosel’skiĭ’s fixed point theorem in a cone; weak Harnack inequality

Paper coordinates

N. Kolun, R. Precup, Energy-based localization of positive solutions for stationary Kirchhoff-type equations and systems, Georgian Math. J.,



About this paper

Publisher Name
Print ISSN


Online ISSN


google scholar link

[1] G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal. 73 (2010), no. 7, 1952–1965. 10.1016/ in Google Scholar

[2] G. Autuori, P. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal. 196 (2010), no. 2, 489–516. 10.1007/s00205-009-0241-xSearch in Google Scholar

[3] H. Brezis, Functional Aalysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. 10.1007/978-0-387-70914-7Search in Google Scholar

[4] G. F. Carrier, On the non-linear vibration problem of the elastic string, Quart. Appl. Math. 3 (1945), 157–165. 10.1090/qam/12351Search in Google Scholar

[5] G. F. Carrier, A note on the vibrating string, Quart. Appl. Math. 7 (1949), 97–101. 10.1090/qam/28511Search in Google Scholar

[6] F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x) -polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (2011), no. 17, 5962–5974. 10.1016/ in Google Scholar

[7] M. Dreher, The Kirchhoff equation for the p-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino 64 (2006), no. 2, 217–238. Search in Google Scholar

[8] G. Kirchhoff, Vorlesungen über Mechanik, Teubner, Leipzig, 1883. Search in Google Scholar

[9] M. A. Krasnosel’skiĭ, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, 1964. Search in Google Scholar

[10] J.-L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary developments in continuum mechanics and partial differential equations, North-Holland Math. Stud. 30, North-Holland, Amsterdam (1978), 284–346. 10.1016/S0304-0208(08)70870-3Search in Google Scholar

[11] T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal. 63 (2005), JIss5–7, e1967–e1977. 10.1016/ in Google Scholar

[12] T. F. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003), no. 2, 243–248. 10.1016/S0893-9659(03)80038-1Search in Google Scholar

[13] R. Narashima, Nonlinear vibration of an elastic string, J. Sound Vibration 8 (1968), no. 1, 134–146. 10.1016/0022-460X(68)90200-9Search in Google Scholar

[14] D. W. Oplinger, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Amer. 32 (1960), 1529–1538. 10.1121/1.1907948Search in Google Scholar

[15] S. I. Pokhozhaev, A quasilinear hyperbolic Kirchhoff equation, Differ. Uravn. 21 (1985), no. 1, 101–108, 182. Search in Google Scholar

[16] R. Precup, A vector version of Krasnosel’skiĭ’s fixed point theorem in cones and positive periodic solutions of nonlinear systems, J. Fixed Point Theory Appl. 2 (2007), no. 1, 141–151. 10.1007/s11784-007-0027-4Search in Google Scholar

[17] R. Precup, Linear and Semilinear Partial Differential Equations. An Introduction, De Gruyter Textbook, Walter de Gruyter, Berlin, 2013. 10.1515/9783110269055Search in Google Scholar

[18] R. Precup, On a bounded critical point theorem of Schechter, Stud. Univ. Babeş-Bolyai Math. 58 (2013), no. 1, 87–95. Search in Google Scholar

[19] R. Precup, P. Pucci and C. Varga, Energy-based localization and multiplicity of radially symmetric states for the stationary p-Laplace diffusion, Complex Var. Elliptic Equ. 65 (2020), no. 7, 1198–1209. 10.1080/17476933.2019.1574774Search in Google Scholar

[20] R. Precup and A. Stan, Stationary Kirchhoff equations and systems with reaction terms, AIMS Math. 7 (2022), no. 8, 15258–15281. 10.3934/math.2022836Search in Google Scholar

[21] P. Pucci and V. D. Rădulescu, Progress in nonlinear Kirchhoff problems [Editorial], Nonlinear Anal. 186 (2019), 1–5. 10.1016/ in Google Scholar

[22] P. Pucci and S. Saldi, Asymptotic stability for nonlinear damped Kirchhoff systems involving the fractional p-Laplacian operator, J. Differential Equations 263 (2017), no. 5, 2375–2418. 10.1016/j.jde.2017.02.039Search in Google Scholar

[23] B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46 (2010), no. 4, 543–549. 10.1007/s10898-009-9438-7Search in Google Scholar

[24] C. F. Vasconcellos, On a nonlinear stationary problem in unbounded domains, Rev. Mat. Univ. Complut. Madrid 5 (1992), no. 2–3, 309–318. 10.5209/rev_REMA.1992.v5.n2.17919Search in Google Scholar


Related Posts