Applications of Mathematics, Vol. 68, No. 4, pp. 425-439, 2023
A new approach to solving a quasilinear boundary value problem with $p$-Laplacian using optimization
Michaela Bailová, Jiří Bouchala
Received August 21, 2022. Published online June 9, 2023.
Abstract: We present a novel approach to solving a specific type of quasilinear boundary value problem with $p$-Laplacian that can be considered an alternative to the classic approach based on the mountain pass theorem. We introduce a new way of proving the existence of nontrivial weak solutions. We show that the nontrivial solutions of the problem are related to critical points of a certain functional different from the energy functional, and some solutions correspond to its minimum. This idea is new even for $p=2$. We present an algorithm based on the introduced theory and apply it to the given problem. The algorithm is illustrated by numerical experiments and compared with the classic approach.
Keywords: $p$-Laplacian operator; quasilinear elliptic PDE; critical point and value; optimization algorithm; gradient method
References: [1] A. Ambrosetti, P. H. Rabinowitz: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349-381. DOI 10.1016/0022-1236(73)90051-7 | MR 0370183 | Zbl 0273.49063
[2] M. Bailová, J. Bouchala: A mountain pass algorithm for quasilinear boundary value problem with $p$-Laplacian. Math. Comput. Simul. 189 (2021), 291-304. DOI 10.1016/j.matcom.2021.03.006 | MR 4297869 | Zbl 07431491
[3] V. Barutello, S. Terracini: A bisection algorithm for the numerical mountain pass. NoDEA, Nonlinear Differ. Equ. Appl. 14 (2007), 527-539. DOI 10.1007/s00030-007-4065-9 | MR 2374198 | Zbl 1141.46036
[4] J. Bouchala, P. Drábek: Strong resonance for some quasilinear elliptic equations. J. Math. Anal. Appl. 245 (2000), 7-19. DOI 10.1006/jmaa.2000.6713 | MR 1756573 | Zbl 0970.35062
[5] G. Chen, J. Zhou, W.-M. Ni: Algorithms and visualization for solutions of nonlinear elliptic equations. Int. J. Bifurcation Chaos Appl. Sci. Eng. 10 (2000), 1565-1612. DOI 10.1142/S0218127400001006 | MR 1780923 | Zbl 1090.65549
[6] Y. S. Choi, P. J. McKenna: A mountain pass method for the numerical solution of semilinear elliptic problems. Nonlinear Anal., Theory Methods Appl. 20 (1993), 417-437. DOI 10.1016/0362-546X(93)90147-K | MR 1206432 | Zbl 0779.35032
[7] Z. Ding, D. Costa, G. Chen: A high-linking algorithm for sign-changing solutions of semilinear elliptic equations. Nonlinear Anal., Theory Methods Appl. 38 (1999), 151-172. DOI 10.1016/S0362-546X(98)00086-8 | MR 1697049 | Zbl 0941.35023
[8] P. Drábek, A. Kufner, F. Nicolosi: Quasilinear Elliptic Equations with Degenerations and Singularities. de Gruyter Series in Nonlinear Analysis and Applications 5. Walter de Gruyter, Berlin (1997). DOI 10.1515/9783110804775 | MR 1460729 | Zbl 0894.35002
[9] J. Horák: Constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems. Numer. Math. 98 (2004), 251-276. DOI 10.1007/s00211-004-0544-7 | MR 2092742 | Zbl 1058.65129
[10] J. Horák, G. Holubová, P. Nečesal (eds.): Proceedings of Seminar in Differential Equations: Deštné v Orlických horách, May 21-25, 2012. Volume I. Mountain Pass and Its Applications in Analysis and Numerics. University of West Bohemia, Pilsen (2012) .
[11] Y. Q. Huang, R. Li, W. Liu: Preconditioned descent algorithms for $p$-Laplacian. J. Sci. Comput. 32 (2007), 343-371. DOI 10.1007/s10915-007-9134-z | MR 2320575 | Zbl 1134.65079
[12] R. Kippenhahn, A. Weigert, A. Weiss: Stellar Structure and Evolution. Astronomy and Astrophysics Library. Springer, Berlin (2012). DOI 10.1007/978-3-642-30304-3
[13] Y. Li, J. Zhou: A minimax method for finding multiple critical points and its applications to semilinear PDEs. SIAM J. Sci. Comput. 23 (2001), 840-865. DOI 10.1137/S1064827599365641 | MR 1860967 | Zbl 1002.35004
[14] Q. Shen: A meshless scaling iterative algorithm based on compactly supported radial basis functions for the numerical solution of Lane-Emden-Fowler equation. Numer. Methods Partial Differ. Equations 28 (2012), 554-572. DOI 10.1002/num.20635 | MR 2879794 | Zbl 1457.65232
[15] M. Struwe: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge 34. Springer, Berlin (2000). DOI 10.1007/978-3-662-04194-9 | MR 1736116 | Zbl 0939.49001
[16] N. Tacheny, C. Troestler: A mountain pass algorithm with projector. J. Comput. Appl. Math. 236 (2012), 2025-2036. DOI 10.1016/j.cam.2011.11.011 | MR 2863532 | Zbl 1245.65075
[17] M. Willem: Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications 24. Birkhäuser, Boston (1996). DOI 10.1007/978-1-4612-4146-1 | MR 1400007 | Zbl 0856.49001
[18] X. Yao, J. Zhou: A minimax method for finding multiple critical points in Banach spaces and its application to quasi-linear elliptic PDE. SIAM J. Sci. Comput. 26 (2005), 1796-1809. DOI 10.1137/S1064827503430503 | MR 2142597 | Zbl 1078.58009
Affiliations: Michaela Bailová (corresponding author), Jiří Bouchala, VŠB - Technical University of Ostrava, Faculty of Electrical Engineering and Computer Science, Department of Applied Mathematics, 17. listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic, e-mail: michaela.bailova@vsb.cz, jiri.bouchala@vsb.cz
