Abstract
In this paper, we deal with the Schrödinger’s problems, in the first part we study the theoretical side, we show the existence of at least three weak solutions, our main tools are based on variational inequalities, more precisely, using the three critical points theorem due to Ricceri, existence and multiplicity results are established. In the second part, we are interested in the application side, more exactly, we examine some computational problems on the discretization of finite elements of the p(x)-Laplacian, we propose a quasi-Newton minimization approach for the solution, our numerical tests show that these algorithms are able to resolve the problems with p(x)-Laplacian, for different values of p(x).
Similar content being viewed by others
References
E. Acerbi and G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal., 156 (2001), 121–140.
G. A. Afrouzi and S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the (p1, p2, …, pN)-Laplacian, Nonlinear Anal., (2009), 135–143.
S. Babaie-Kafaki and R. Ghanbari, A hybridization of the polak-ribière-polyak and fletcher-reeves conjugate gradient methods, Numer. Algorithms, (2015), 481–495.
J. W. Barrett and W. B. Liu, Finite element approximation of the p-Laplacian, Math. Comp., 61(204) (1993), 523–537.
R. Bermejo and J.-A. Infante, A multigrid algorithm for the p-Laplacian, SIAM J. Sci. Comput., 21(5) (2000), 1774–1789.
R. J. Biezuner, J. Brown, G. Ercole, and E. M. Martins, Computing the first Eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions, J. Sci. Comput., 52(1) (2012), 180–201.
D. Breit, L. Diening, and S. Schwarzacher, Finite element approximation of the p(x)-Laplacian, SIAM J. Numer. Anal., 53(1) (2015), 551–572.
M. Caliari and S. Zuccher, The inverse power method for the p(x)-Laplacian problem, J. Sci. Comput., 65(2), (2015), 698–714.
M. Caliari and S. Zuccher, Quasi-Newton minimization for the p(x)-Laplacian problem, J. Computational and Applied Mathematics, 309 (2017), 122–131.
L. M. Del Pezzo and S. Martínez, Order of convergence of the finite element method for the p(x)-Laplacian, IMA J. Numer. Anal., 35(4) (2015), 1864–1887.
L. M. Del Pezzo, A. L. Lombardi, and S. Martínez, Interior penalty discontinuous Galerkin FEM for the p(x)-Laplacian, SIAM J. Numer. Anal., 50(5) (2012), 2497–2521.
J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, In: Classics in Applied Mathematics, vol. SIAM, Philadelphia, PA, USA, 1996.
A. Djellit and S. Tas, Existence of solutions for a class of elliptic systems in ℝN involving the p-Laplacian, Electron. J. Differential Equations, (2003), 1–8.
A. Djellit, Z. Youbi, and S. Tas, Existence of solutions for elliptic systems in ℝN involving the p(x)-Laplacian, ejde, Vol. 2012, No. 131, pp. 1–10, 2012.
D. E. Edmunds and J. Rakosnik, Sobolev embeddings with variable exponent, Studia Mathematica, 143(3) (2000), 267–293.
X. L. Fan, J. S. Shen, and D. Zhao, Sobolev embedding theorems for spaces Wk,p(x) (Ω), J. Math. Anal. Appl., 262 (2001), 749–760.
X.-L. Fan and D. Zhao, On the spaces Lp(x) and Wm,p(x), J. Math. Anal. Appl., 263(2) (2001), 424–446.
X.-L. Fan, Q. Zhang, and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302(2) (2005), 306–317.
X.-L. Fan, Y. Z. Zhao, and D. Zhao, compact imbedding theorems with symmetry of Strauss-Lions type for the space W1,p(x) (Ω), J. Math. Anal. Appl., 255 (2001), 333–348.
A. El Hamidi, Existence results to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl., 300(1) (2004), 30–42.
F. Hecht, New development in FreeFem++, J. Numer. Math., 20(3–4) (2012), 251–265.
A. Hirn, Finite element approximation of singular power-law systems, Math. Comp., 82(283) (2013), 1247–1268.
Y. Q. Huang, R. Li, and W. Liu, Preconditioned descent algorithms for p-Laplacian, J. Sci. Comput., 32(2) (2007), 343–371.
T. Iwaniec, J. J. Manfredi, Regularity of p-harmonic functions on the plane, Rev. Mat. Iberoamericana, 5 (1989), 1–19.
C. T. Kelley, Iterative methods for optimization, In: Frontiers in Applied Mathematics, SIAM, 18, Philadelphia, 1999.
S. Kichenassamy and L. Veron, Singular solutions of the p-Laplace equation, Math. Ann., 275 (1985.), 599–615.
A. Kristaly, Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains, Proc. Edinb. Math. Soc. (2), 48(2) (2005), 465–477.
P. L. Lions, The concentration-compactness principale in the calculus of variations. The locally compact case, Part 2, Ann. Inst. H. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 223–283.
S. Ogras, R. A. Mashiyev, M. Avci, and Z. Yucedag, Existence of Solutions for a class of Elliptic Systems in ℝN Involving the (p (x), q (x))-Laplacian, Journal of Inequalities and Applications, 2008 (2008), Article ID 612938, 16 pages.
B. Ricceri, Existence of three solutions for a class of elliptic Eigenvalue problems, Math. Comput. Modelling, 32 (2000), 1485–1494.
B. Ricceri, A three critical points theorem revisited, Nonlinear Analysis, 70 (2009), 3084–3089.
M. Ruzicka, Electro-rheological fluids: Modeling and mathematical theory, In Lecture Notes in Math., 1784 (2000), Springer-Verlag, Berlin.
F. D. Thelin, Local regularity properties for the solutions of a nonlinear partial differential equation, Nonlinear Anal., 6 (1982), 839–844.
H. Yin and Z. Yang, Three solutions for a class of quasilinear elliptic systems involving the p(x)-Laplace operator, Boundary Value Problems a Springer Open Journal, 2012.
X. Xu and Y. An, Existence and multiplicity of solutions for elliptic systems with nonstandard growth condition in ℝN, Mathematische Nachrichten, 268(1) (2004), 31–43.
E. Zeidler, Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators, Springer, New York, 1990.
W. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv., 29 (1987), 33–66.
G. Zhou, Y. Huang, and C. Feng, Preconditioned hybrid conjugate gradient algorithm for p-Laplacian, Int. J. Numer. Anal. Model, 2(Suppl.) (2005), 123–130.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Chergui, T., Tas, S. Existence, multiplicity and numerical examples for Schrödinger systems with nonstandard p(x)-growth conditions. Indian J Pure Appl Math 51, 413–437 (2020). https://doi.org/10.1007/s13226-020-0408-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-020-0408-6
Key words
- Nonlinear boundary value problems
- variable exponent sobolev space
- multiplicity results
- quasi-Newton minimization