Optimizing solutions with competing anisotropic (p, q)-Laplacian in hemivariational inequalities

Year 2024, Volume: 7 Issue: 4, 150 - 159, 15.12.2024

Dumitru Motreanu Abdolrahman Razani

https://doi.org/10.33205/cma.1566388

Abstract

For differential inclusions and hemivariational inequalities driven by anisotropic differential operators, we establish the existence of generalized variational solutions and weak solutions. The main novelty consists in allowing that the driving operators might not satisfy any ellipticity condition, which is achieved for the first time in the anisotropic and nonsmooth context. The approach is based on a finite dimensional approximation process.

Keywords

Differential inclusion, Hermivariational inequality, Anisotropic p-laplacian, competing operators, generalized variational solution, weak solution

Project Number

0

References

• M. Allalou, M. El Ouaarabi and A. Raji: On a class of nonhomogeneous anisotropic elliptic problem with variable exponents, Rend. Circ. Mat. Palermo, II. Ser (2024).
• M. Bohner, G. Caristi, A. Ghobadi and Sh. Heidarkhani: Three solutions for discrete anisotropic Kirchhoff-type problems, Demonstr. Math., 56 (1) (2023), Article ID: 20220209.
• G. Bonanno, G. D’Aguì and A. Sciammetta: Multiple solutions for a class of anisotropic −→p -Laplacian problems, Bound. Value Probl., 2023 (2023), Article ID: 89.
• B. Brandolini, F. Cîrstea: Anisotropic elliptic equations with gradient-dependent lower order terms and L1 data, Math. Eng., 5 (4) (2023), 1–33.
• B. Brandolini, F. Cîrstea: Boundedness of solutions to singular anisotropic elliptic equations, Discrete Contin. Dyn. Syst. Ser. S, 17 (4) (2024), 1545–1561.
• H. Brezis: Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York (2011).
• A. Cernea: On the solutions of a coupled system of proportional fractional differential inclusions of Hilfer type, Modern Math. Methods, 2 (2) (2024), 80–89.
• K. C. Chang: Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102–129.
• F. H. Clarke: Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York, USA (1983).
• I. Fragalà, F. Gazzola and B. Kawohl: Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 21 (5) (2004), 715–734.
• L. Gambera, S. A. Marano and D. Motreanu: Quasi-linear Dirichlet systems with competing operators and convection, J. Math. Anal. Appl., 530 (2024), Article ID: 127718.
• Z. Liu, R. Livrea, D. Motreanu and S. Zeng: Variational differential inclusions without ellipticity condition, Electron. J. Qual. Theory Differ. Equ., 43 (2020), 1–17.
• M. Mih˘ailescu, P. Pucci and V. D. R˘adulescu: Eigenvalue problems for anisotropic quasilinear elliptic equations with varaible exponent, J. Math. Anal. Appl., 340 (2008), 687–698.
• D. Motreanu: Quasilinear Dirichlet problems with competing operators and convection, Open Math., 18 (2020), 1510–1517.
• D. Motreanu: Systems of hemivariational inclusions with competing operators, Mathematics, 12 (11) (2024), Article ID: 1766.
• D. Motreanu: Hemivariational inequalities with competing operators, Commun. Nonlinear Sci. Numer. Simulat., 130 (2024), Article ID: 107741.
• D. Motreanu, A. Razani: Competing anisotropic and Finsler (p, q)-Laplacian problems, Bound. Value Probl., 2024 (2024), Article ID: 39.
• D. Motreanu, E. Tornatore: Dirichlet problems with anisotropic principal part involving unbounded coefficients, Electron. J. Differential Equations, 2024 (11) (2024), 1–13.
• A. Razani: Nonstandard competing anisotropic (p, q)-Laplacians with convolution, Bound. Value Probl., 2022 (2022), Article ID: 87.
• A. Razani: Entire weak solutions for an anisotropic equation in the Heisenberg group, Proc. Amer. Math. Soc., 151 (11) (2023), 4771–4779.
• A. Razani: Competing Kohn-Spencer Laplacian systems with convection in non-isotropic Folland-Stein space, Complex Var. Elliptic Equ., (2024), 1–14. DOI: 10.1080/17476933.2024.2337868
• A. Razani, G. S. Costa and G. M. Figueiredo: A positive solution for a weighted anisotropic p-Laplace equation involving vanishing potential, Mediterr. J. Math., 21 (2024), Article ID: 59.
• A. Razani, G. M. Figueiredo: Degenerated and competing anisotropic (p, q)-Laplacians with weights, Appl. Anal., 102 (2023), 4471–4488.
• A. Razani, G. M. Figueiredo: A positive solution for an anisotropic (p, q)-Laplacian, Discrete Contin. Dyn. Syst. Ser. S, 16 (6) (2023), 1629–1643.
• A. Razani, G. M. Figueiredo: Infinitely many solutions for an anisotropic differential inclusion on unbounded domains, Electron. J. Qual. Theory Differ. Equ., 33 (2024), 1–17.
• A. Razani, E. Tornatore: Solutions for nonhomogeneous degenerate quasilinear anisotropic problems, Constr. Math. Anal., 7 (3) (2024), 134–149.