Abstract
In this paper, we investigate two optimization problems related to a quasilinear elliptic equation with p-Laplacian, logistic-type growth rate function such that the admissible set is a class of rearrangements of a fixed function. Under some suitable assumptions, we prove existence and representation of the maximizers and existence, uniqueness and representation of the minimizer. Also, when the domain of the equation is a ball, we show that the maximizer is unique and symmetric.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Adams, Sobolev spaces, Academic Press, New York (1975).
C. D. Aliprantis and O. Burkinshaw, Principles of real analysis, Second Edition, Academic Press (1990).
N. Amiri and M. Zivari-Rezapour, Maximization and minimization problems related to a p-Laplacian equation on a multiply conected domain, Taiwanese Journal of Mathematics, 19(1) (2015), 243–252.
G. R. Burton, Rearrangements of Functions, Maximization of Convex Functionals, and Vortex Rings, Math. Ann., 276 (1987), 225–253.
G. R. Burton, Variational problems on classes of rearrangements and multiple configurations for steady vortices, Ann. Inst. H. Poincaré Anal. Non linéaire, 6 (1989), 295–319.
G. R. Burton, Rearrangements of functions, saddle points and uncountable families of steady configurations for a vortex, Acta Math., 163 (1989), 291–309.
F. Cuccu, B. Emamizadeh, and G. Porru, Optimization of the first eigenvalue in problems involving the p-Laplacian, Proc. Amer. Math. Soc., 137(5) (2009), 1677–1687.
F. Cuccu, B. Emamizadeh, and G. Porru, Optimization problems for an elastic plate, J. Math. Phys., 47(8) (2006), 082901, 12 pp.
F. Cuccu, B. Emamizadeh, and G. Porru, Nonlinear elastic membranes involving the p-Laplacian operator, Electron. J. Differential Equations, 49 (2006) 10 pp.
F. Cuccu, G. Porru, and S. Sakaguchi, Optimization problems on general classes of rearrangements, Nonlinear Analysis, 74 (2011), 5554–5565.
F. Cuccu, G. Porru, and A. Vitolo, Optimization of the energy integral in two classes of rearrangements, Nonlinear Stud., 17(1) (2010), 23–35.
L. M. Del Pezzo and J. Fernández Bonder, Some optimization problems for p-Laplacian type equations, Appl. Math. Optim., 59(3) (2009), 365–381.
L. M. Del Pezzo and J. Fernández Bonder, Remarks on an optimization problem for the p-Laplacian, Appl. Math. Lett., 23(2) (2010), 188–192.
P. Drabek and J. Hernandez, Existence And Uniqueness Of Positive Solutions For Some Quasilinear Elliptic Problems, Nonlinear Analysis, 44 (2001), 189–204.
B. Emamizadeh and M. Zivari-Rezapour, Optimization of the principal eigenvalue of the pseudo p-Laplacian operator with Robin boundary conditions, International Journal of Mathematics, 23(12) (2012), 1250127, 17 pp.
B. Emamizadeh and M. Zivari-Rezapour, Rearrangements and minimization of the principal eigenvalue of a nonlinear Steklov problem, Nonlinear Anal., 74(16) (2011), 5697–5704.
B. Emamizadeh and M. Zivari-Rezapour, Rearrangement optimization for some elliptic equations, J. Optim. Theory Appl., 135(3) (2007), 367–379.
B. Emamizadeh and J. V. Prajapat, Maximax and minimax rearrangement optimization problems, Optim. Lett., 5(4) (2011), 647–664.
B. Kawohl, Rearrangements and convexity of level sets in PDE, Lectures Notes in Mathematics, 1150, Berlin (1985).
M. Zivari-Rezapour, Maximax rearrangement optimization related to a homogeneous Dirichlet problem, Arab. J. Math., 2(4) (2013), 427–433.
Acknowledgement
We are grateful to the Research Council of Shahid Chamran University of Ahvaz for financial support (SCU.MM98.441).
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Amiri, N., Zivari-Rezapour, M. Maximization and minimization problems related to an equation with the p-Laplacian. Indian J Pure Appl Math 51, 777–788 (2020). https://doi.org/10.1007/s13226-020-0430-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-020-0430-8