The dirichlet problem for a class of anisotropic Schrödinger-Kirchhoff-type equations with critical exponent
Abstract
In this paper, our focus lies in addressing the Dirichlet problem associated with a specific class of critical anisotropic elliptic equations of Schrödinger-Kirchhoff type. These equations incorporate variable exponents and a real positive parameter. Our objective is to establish the existence of at least one solution to this problem.
Keyword : Schrodinger-Kirchhoff-type problems, Dirichlet boundary conditions, p(x)-Laplacian, Anisotropic variable exponent Sobolev spaces, Concentration-compactness principle, parameter
How to Cite
Chems Eddine, N., Nguyen, A. T., & Ragusa, M. A. (2024). The dirichlet problem for a class of anisotropic Schrödinger-Kirchhoff-type equations with critical exponent. Mathematical Modelling and Analysis, 29(2), 254–267. https://doi.org/10.3846/mma.2024.19006
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
M.J. Ablowitz, B. Prinari and A.D. Trubatch. Discrete and continuous nonlinear Schrödinger systems, volume 302. Cambridge University Press, 2004.
C.O. Alves and J.L.P. Barreiro. Existence and multiplicity of solutions for a p(x)-Laplacian equation with critical growth. Journal of Mathematical Analysis and Applications, 403(1):143–154, 2013. https://doi.org/10.1016/j.jmaa.2013.02.025
C.O. Alves, F. Corrêa and G.M. Figueiredo. On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl, 2(3):409–417, 2010. https://doi.org/10.7153/dea-02-25
J.F. Bonder, N. Saintier and A. Silva. The concentration-compactness principle for fractional order sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem. Nonlinear Differential Equations and Applications NoDEA, 25:1–25, 2018. https://doi.org/10.1007/s00030-018-0543-5
H. Brézis and L. Nirenberg. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Communications on pure and applied mathematics, 36(4):437–477, 1983. https://doi.org/10.1002/cpa.3160360405
N. Chems Eddine. Existence and multiplicity of solutions for Kirchhoff-type potential systems with variable critical growth exponent. Applicable Analysis, pp. 1–21, 2021. https://doi.org/10.1080/00036811.2021.1979223
N. Chems Eddine. Multiple solutions for a class of generalized critical noncooperative Schrödinger systems in Rn. Results in Mathematics, 78(6):226, 2023. https://doi.org/10.1007/s00025-023-02005-2
N. Chems Eddine and M.A. Ragusa. Generalized critical Kirchhoff-type potential systems with Neumann boundary conditions. Applicable Analysis, 101(11):3958–3988, 2022. https://doi.org/10.1080/00036811.2022.2057305
N. Chems Eddine, M.A. Ragusa and D.D. Repovš. On the concentrationcompactness principle for anisotropic variable exponent Sobolev spaces and applications. Fractional Calculus and Applied Analysis, 2024.
N. Chems Eddine and D.D. Repovš. The Neumann problem for a class of generalized Kirchhoff-type potential systems. Boundary Value Problems, 2023(1):1–33, 2023. https://doi.org/10.1186/s13661-023-01705-6
E. DiBenedetto. Degenerate parabolic equations. Springer Science & Business Media, 1993. https://doi.org/10.1007/978-1-4612-0895-2
X. Fan. Anisotropic variable exponent Sobolev spaces and p(x)-Laplacian equations. Complex Variables and Elliptic Equations, 56(7-9):623–642, 2011. https://doi.org/10.1080/17476931003728412
X. Fan and D. Zhao. On the spaces Lp(x)(ω) and Wm,p(x)(ω). Journal of Mathematical Analysis and applications, 263(2):424–446, 2001. https://doi.org/10.1006/jmaa.2000.7617
A. Fiscella and E. Valdinoci. A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Analysis: Theory, Methods & Applications, 94:156– 170, 2014. https://doi.org/10.1016/j.na.2013.08.011
Y. Fu. The principle of concentration compactness in Lp(x) spaces and its application. Nonlinear Analysis: Theory, Methods & Applications, 71(5–6):1876–1892, 2009. https://doi.org/10.1016/j.na.2009.01.023
Y. Fu and X. Zhang. Multiple solutions for a class of p(x)-Laplacian equations in involving the critical exponent. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466(2118):1667–1686, 2010. https://doi.org/10.1098/rspa.2009.0463
A. Hamydy, M. Massar and N. Tsouli. Existence of solutions for p-Kirchhoff type problems with critical exponent. Electron. J. Differential Equations, 105:1–8, 2011.
G. Kirchhoff. Vorlesungen über Mechanik, volume 1. Teubner, Leipzig, 1883.
P. Lions. The concentration-compactness principle in the calculus of variation, the limit case, part 1. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 1(2):109–145, 1984. https://doi.org/10.1016/S0294-1449(16)30428-0
V.D. Radulescu and D.D. Repovs. Partial differential equations with variable exponents: variational methods and qualitative analysis, volume 9. CRC Press, 2015.
D. Repovš. Stationary waves of Schrödinger-type equations with variable exponent. Analysis and Applications, 13(06):645–661, 2015. https://doi.org/10.1142/S0219530514500420
E. Schrödinger. An undulatory theory of the mechanics of atoms and molecules. Physical Review, 28(6):1049, 1926.
C. Sulem and P.-L. Sulem. The nonlinear Schr¨odinger equation: self-focusing and wave collapse, volume 139. Springer Science & Business Media, 2007. https://doi.org/10.1007/b98958
C.O. Alves and J.L.P. Barreiro. Existence and multiplicity of solutions for a p(x)-Laplacian equation with critical growth. Journal of Mathematical Analysis and Applications, 403(1):143–154, 2013. https://doi.org/10.1016/j.jmaa.2013.02.025
C.O. Alves, F. Corrêa and G.M. Figueiredo. On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl, 2(3):409–417, 2010. https://doi.org/10.7153/dea-02-25
J.F. Bonder, N. Saintier and A. Silva. The concentration-compactness principle for fractional order sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem. Nonlinear Differential Equations and Applications NoDEA, 25:1–25, 2018. https://doi.org/10.1007/s00030-018-0543-5
H. Brézis and L. Nirenberg. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Communications on pure and applied mathematics, 36(4):437–477, 1983. https://doi.org/10.1002/cpa.3160360405
N. Chems Eddine. Existence and multiplicity of solutions for Kirchhoff-type potential systems with variable critical growth exponent. Applicable Analysis, pp. 1–21, 2021. https://doi.org/10.1080/00036811.2021.1979223
N. Chems Eddine. Multiple solutions for a class of generalized critical noncooperative Schrödinger systems in Rn. Results in Mathematics, 78(6):226, 2023. https://doi.org/10.1007/s00025-023-02005-2
N. Chems Eddine and M.A. Ragusa. Generalized critical Kirchhoff-type potential systems with Neumann boundary conditions. Applicable Analysis, 101(11):3958–3988, 2022. https://doi.org/10.1080/00036811.2022.2057305
N. Chems Eddine, M.A. Ragusa and D.D. Repovš. On the concentrationcompactness principle for anisotropic variable exponent Sobolev spaces and applications. Fractional Calculus and Applied Analysis, 2024.
N. Chems Eddine and D.D. Repovš. The Neumann problem for a class of generalized Kirchhoff-type potential systems. Boundary Value Problems, 2023(1):1–33, 2023. https://doi.org/10.1186/s13661-023-01705-6
E. DiBenedetto. Degenerate parabolic equations. Springer Science & Business Media, 1993. https://doi.org/10.1007/978-1-4612-0895-2
X. Fan. Anisotropic variable exponent Sobolev spaces and p(x)-Laplacian equations. Complex Variables and Elliptic Equations, 56(7-9):623–642, 2011. https://doi.org/10.1080/17476931003728412
X. Fan and D. Zhao. On the spaces Lp(x)(ω) and Wm,p(x)(ω). Journal of Mathematical Analysis and applications, 263(2):424–446, 2001. https://doi.org/10.1006/jmaa.2000.7617
A. Fiscella and E. Valdinoci. A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Analysis: Theory, Methods & Applications, 94:156– 170, 2014. https://doi.org/10.1016/j.na.2013.08.011
Y. Fu. The principle of concentration compactness in Lp(x) spaces and its application. Nonlinear Analysis: Theory, Methods & Applications, 71(5–6):1876–1892, 2009. https://doi.org/10.1016/j.na.2009.01.023
Y. Fu and X. Zhang. Multiple solutions for a class of p(x)-Laplacian equations in involving the critical exponent. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466(2118):1667–1686, 2010. https://doi.org/10.1098/rspa.2009.0463
A. Hamydy, M. Massar and N. Tsouli. Existence of solutions for p-Kirchhoff type problems with critical exponent. Electron. J. Differential Equations, 105:1–8, 2011.
G. Kirchhoff. Vorlesungen über Mechanik, volume 1. Teubner, Leipzig, 1883.
P. Lions. The concentration-compactness principle in the calculus of variation, the limit case, part 1. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 1(2):109–145, 1984. https://doi.org/10.1016/S0294-1449(16)30428-0
V.D. Radulescu and D.D. Repovs. Partial differential equations with variable exponents: variational methods and qualitative analysis, volume 9. CRC Press, 2015.
D. Repovš. Stationary waves of Schrödinger-type equations with variable exponent. Analysis and Applications, 13(06):645–661, 2015. https://doi.org/10.1142/S0219530514500420
E. Schrödinger. An undulatory theory of the mechanics of atoms and molecules. Physical Review, 28(6):1049, 1926.
C. Sulem and P.-L. Sulem. The nonlinear Schr¨odinger equation: self-focusing and wave collapse, volume 139. Springer Science & Business Media, 2007. https://doi.org/10.1007/b98958