Posts by Călin-Ioan Gheorghiu

Abstract

The paper deals with existence, localization and multiplicity of radial positive solutions in the annulus or the ball, for the Neumann problem involving a general φ-Laplace operator. Our results apply in particular to the classical Laplacian and to the mean curvature operators in the Euclidean and Minkowski spaces. Numerical experiments with the MATLAB object-oriented package Chebfun are performed to obtain numerical solutions for some concrete equations.

Authors

Radu Precup
Institute of Advanced Studies in Science and Technology, Babeş-Bolyai, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Călin-Ioan Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Keywords

Neumann boundary value problem; φ-Laplace operator; Radial solution; Positive solution; Fixed point index; Harnack type inequality; Numerical solution

Paper coordinates

R. Precup, C.-I. Gheorghiu, Theory and computation of radial solutions for Neumann problems with φ-Laplacian, Qualitative Theory of Dynamical Systems, 23 (, art. no. 107, https://doi.org/10.1007/s12346-024-00963-8

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About this paper

Journal

Qualitative Theory of Dynamical Systems

Publisher Name

Springer International Publishing

Print ISSN
1662-3592

 

Online ISSN

1575-5460

google scholar link

[1] Bartnik, R., Simon, L.: Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys. 87, 131–152 (1982). https://doi.org/10.1007/BF01211061, Article ADS MathSciNet Google Scholar
[2] Benedikt, J., Girg, P., Kotrla, L., Takac, P.: Origin of the φ-Laplacian and A. Missbach. Electron. J. Differ. Equ. 2018(16), 1–17 (2018). http://ejde.math.txstate.edu or http://ejde.math.unt.edu
[3] Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces. Math. Nachr. 283, 379–391 (2010). https://doi.org/10.1002/mana.200910083, Article MathSciNet Google Scholar
[4] Bonheure, D., Noris, B., Weth, T.: Increasing radial solutions for Neumann problems without growth restrictions. Ann. Inst. H. Poincaré Anal. Non Linéaire 29, 573–588 (2012). https://doi.org/10.1016/j.anihpc.2012.02.002, Article ADS MathSciNet Google Scholar
[5] Bonheure, D., Serra, E., Tilli, P.: Radial positive solutions of elliptic systems with Neumann boundary conditions. J. Funct. Anal. 265, 375–398 (2013). https://doi.org/10.1016/j.jfa.2013.05.027, Article MathSciNet Google Scholar
[6] Boscaggin, A., Colasuonno, F., Noris, B.: Multiple positive solutions for a class of φ-Laplacian Neumann problems without growth conditions. ESAIM Control Optim. Calc. Var. 24, 1625–1644 (2018). https://doi.org/10.1051/cocv/2017074, Article MathSciNet Google Scholar
[7] Boscaggin, A., Feltrin, G.: Pairs of positive radial solutions for a Minkowski-curvature Neumann problem with indefinite weight. Nonlinear Anal. 196, 111807 (2020). https://doi.org/10.1016/j.na.2020.111807, Article MathSciNet Google Scholar
[8] Colasuonno, F., Noris, B.: Radial positive solutions for Laplacian supercritical Neumann problems. Bruno Pini Math. Anal. Semin. 8 (1), 55–72 (2017).
https://doi.org/10.6092/issn.2240-2829/7797,

 

[9] Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985) Book Google Scholar
[10] Gheorghiu, C.I.: A third-order nonlinear BVP on the half-line, Chebfun (2020). https://www.chebfun.org/examples/ode-nonlin/GulfStream.html
[11] López-Gómez, J., Omari, P., Rivetti, S.: Positive solutions of a one-dimensional indefinite capillarity-type problem: a variational approach. J. Differ. Equ. 262, 2335–2392 (2017) Article ADS MathSciNet Google Scholar
[12] Ma, R., Chen, T., Wang, H.: Nonconstant radial positive solutions of elliptic systems with Neumann boundary conditions. J. Math. Anal. Appl. 443, 542–565 (2016). https://doi.org/10.1016/j.jmaa.2016.05.038, Article MathSciNet Google Scholar
[13] O’Regan, D., Wang, H.: Positive radial solutions for φ-Laplacian systems. Aequ. Math. 75, 43–50 (2008). https://doi.org/10.1007/s00010-007-2909-3 Article MathSciNet Google Scholar
[14] Precup, R.: Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems. J. Math. Anal. Appl. 352, 48–56 (2009). https://doi.org/10.1016/j.jmaa.2008.01.097, Article MathSciNet Google Scholar
[15] Precup, R., Pucci, P., Varga, C.: Energy-based localization and multiplicity of radially symmetric states for the stationary φ-Laplace diffusion. Complex Var. Ellipt. Equ. 65, 1198–1209 (2020). https://doi.org/10.1080/17476933.2019.1574774 Article MathSciNet Google Scholar
[16] Precup, R., Rodriguez-Lopez, J.: Positive radial solutions for Dirichlet problems via a Harnack type inequality Math. Meth. Appl. Sci. 46(2), 2972–2985 (2023). https://doi.org/10.1002/mma.8682
[17] Trefethen, L.N.: Computing numerically with functions instead of numbers. Math. Comput. Sci. 1, 9–19 (2007). https://doi.org/10.1007/s11786-007-0001-y Article MathSciNet Google Scholar
[18] Trefethen, L.N., Birkisson, A., Driscoll, T.A.: Exploring ODEs. SIAM, Philadelphia (2018) Google Scholar
[19] Wang, H.: On the existence of positive solutions for semilinear elliptic equations in the annulus. J. Differ. Equ. 109, 1–7 (1994). https://doi.org/10.1006/jdeq.1994.1042

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