A Class of Subcritical and Critical Schrödinger–Kirchhoff Equations with Variable Exponents
Abstract
1. Introduction
- (M1)
- , and there exists a constant such that .
- (M2)
- There exists satisfying for each , where .
- (V1)
- . Moreover, there exists such that and .
- (V2)
- satisfies , and for every , .
- (H1)
- There exist and such that
- (H2)
- as uniformly in .
- (H3)
- uniformly in .
- (H4)
- There exists such that
- (H5)
- , where .
2. Preliminaries
- (i)
- there are constants such that ;
- (ii)
- there is a with such that .
3. Proof of Theorem 1
4. Proof of Theorem 2
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Kirchhoff, G. Mechanik; Teubner: Leipzig, Germany, 1883. [Google Scholar]
- Alves, C.O.; Souto, M.A.S. Existence of solutions for a class of nonlinear Schröinger equations with potential vanishing at infinity. J. Differ. Equ. 2013, 254, 1977–1991. [Google Scholar] [CrossRef]
- Wu, Y.; Chen, W. On strongly indefinite Schröinger equations with non-periodic potential. J. Appl. Anal. Comput. 2023, 13, 1–10. [Google Scholar]
- Khoutir, S.; Chen, H.B. Existence of infinitely many high enery solutions for a fractional Schröinger equation in RN. Appl. Math. Lett. 2016, 61, 156–162. [Google Scholar] [CrossRef]
- Pucci, P.; Xiang, M.Q.; Zhang, B.L. Multiple solutions for nonhomogeneous Schröinger-Kirchhoff type equations involving the fractional p-Laplacian in RN. Calc. Var. Partial. Differ. Equ. 2015, 54, 2785–2806. [Google Scholar] [CrossRef]
- Ding, Y.H.; Szulkin, A. Bound states for semilinear Schrödinger equations with sign-changing potential. Calc. Var. Partial. Differ. Equ. 2007, 29, 397–419. [Google Scholar] [CrossRef]
- Nie, J.J. Existence and multiplicity of nontrivial solutions for a class of Schrödinger-Kirchhoff-type equations. J. Math. Anal. Appl. 2014, 417, 65–79. [Google Scholar] [CrossRef]
- Guo, Y.X.; Nie, J.J. Existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger-Kirchhoff-type equations. J. Math. Anal. Appl. 2015, 428, 1054–1069. [Google Scholar] [CrossRef]
- Thin, N.V.; Thuy, P.T. On existence solution for Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in RN. Complex Var. Elliptic Equ. 2019, 64, 461–481. [Google Scholar] [CrossRef]
- Chen, W.; Fu, Z.W.; Wu, Y. Positive ground states for nonlinear Schrödinger-Kirchhoff equations with periodic potential or potential well in R3. Bound. Value Probl. 2022, 97, 1–16. [Google Scholar]
- Jiang, S.; Liu, S.B. Multiple solutions for Schrödinger-Kirchhoff equations with indefinite potential. Appl. Math. Lett. 2022, 124, 107672. [Google Scholar] [CrossRef]
- Azroul, E.; Benkirane, A.; Srati, M. Three solutions for a Schrödinger-Kirchhoff type equation involving nonlocal fractional integro-defferential operators. J. Pseudo-Differ. Oper. Appl. 2020, 11, 1915–1932. [Google Scholar] [CrossRef]
- Cammaroto, F.; Vilasi, L. On a Schrödinger-Kirchhoff-type equation involving the p(x)-Laplace. Nonlinear Anal. 2013, 81, 42–53. [Google Scholar] [CrossRef]
- Xie, W.H.; Chen, H.B. Existence and multiplicity of solutions for p(x)-Laplacian equations in RN. Math. Nachr. 2018, 291, 2476–2488. [Google Scholar] [CrossRef]
- Fan, X.L.; Zhang, Q.H. Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonliear. Anal. 2003, 52, 1843–1852. [Google Scholar] [CrossRef]
- Fan, X.L. Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients. J. Math. Anal. Appl. 2005, 312, 464–477. [Google Scholar] [CrossRef]
- Alves, C.O.; Liu, S.B. On superlinear p(x)-Laplacian equations in RN. Nonlinear Anal. 2010, 73, 2566–2579. [Google Scholar] [CrossRef]
- Pucci, P.; Zhang, Q.H. Existence of entire solutions for a class of variable exponent elliptic equations. J. Differ. Equ. 2014, 257, 1529–1566. [Google Scholar] [CrossRef]
- R<i>a</i>˘dulescu, V.D. Nonlinear elliptic equations with variable exponent:old and new. Nonlinear Anal. 2015, 121, 336–369. [Google Scholar]
- Lee, J.; Kim, J.M.; Kim, Y.H. Existence and multiplicity of solutions for Kirchhoff-Schrödinger type equations involving p(x)-Laplacian on the entire space RN. Nonlinear Anal. Real World Appl. 2019, 45, 620–649. [Google Scholar] [CrossRef]
- Crespo-Blanco, Á.; Gasiński, L.; Harjulehto, P.; Winkert, P. A new class of double phase variable exponent problems: Existence and uniqueness. J. Differ. Equ. 2022, 323, 182–228. [Google Scholar] [CrossRef]
- Ayazoglu, R.; Sarac, Y.; Sener, S.S.; Alisoy, G. Existence and multiplicity of solutions for a Schrödinger-Kirchhoff type equations involving the fractional p(·)-Laplacian operator in RN. Collect. Math. 2021, 72, 129–156. [Google Scholar] [CrossRef]
- Fu, Y.Q.; Zhang, X. Multiple solutions for a class of p(x)-Laplacian equations in RN involving the critical exponent. Proc. R. Soc. Lond. Ser. A. 2010, 466, 1667–1686. [Google Scholar]
- Zhang, Y.P.; Qin, D.D. Existence of solutions for a critical Choquard-Kirchhoff problem with variable exponents. J. Geom. Anal. 2023, 33, 200. [Google Scholar] [CrossRef]
- Ho, K.; Kim, Y.H.; Sim, I. Existence results for Schrödinger p(x)-Laplace equations involving critical growth in RN. Nonlinear Anal. 2019, 182, 20–44. [Google Scholar] [CrossRef]
- Ho, K.; Kim, Y.H.; Lee, J. Schrödinger p(x)-Laplace equations in RN involving indefinite weights and critical growth. J. Math. Phys. 2021, 62, 111506. [Google Scholar] [CrossRef]
- Alves, C.O.; Barreiro, J.L.P. Existence and multiplicity of solutions for a p(x)-Laplacian equation with critical growth. J. Math. Anal. Appl. 2013, 403, 143–154. [Google Scholar] [CrossRef]
- Alves, C.O.; Ferreira, M.C. Nonlinear perturbations of a p(x)-Laplacian equations with critical growth in RN. Math. Nachr. 2014, 287, 849–868. [Google Scholar] [CrossRef]
- Yun, Y.Z.; An, T.Q.; Ye, G.J.; Zuo, J.B. Existence of solutions for asymptotically periodic fractional Schröinger equation with critical growth. Math. Meth. Appl. Sci. 2020, 43, 10081–10097. [Google Scholar] [CrossRef]
- Liang, S.; Pucci, P.; Zhang, B.L. Multiple solutions for critical Choquard-Kirchhoff type equations. Adv. Nonlinear Anal. 2021, 10, 400–419. [Google Scholar] [CrossRef]
- Ji, C.; Rădulescu, V.D. Multi-bump solutions for quasilinear elliptic equations with variable exponents and critical growth in RN. Commun. Contemp. Math. 2021, 23, 41. [Google Scholar] [CrossRef]
- Fiscella, A.; Pucci, P. (p,q) systems with critical terms in RN. Nonlinear Anal. 2018, 177, 454–479. [Google Scholar] [CrossRef]
- Song, Y.Q.; Shi, S.Y. Existnece and multiplicity of solutions for Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity. Appl. Math. Lett. 2019, 92, 170–175. [Google Scholar] [CrossRef]
- Wang, C.; Shang, Y.Y. Existence and multiplicity of solutions for Schröinger equation with inverse square potential and Hardy-Sobolev critical exponent. Nonlinear Anal. Real World Appl. 2019, 46, 525–544. [Google Scholar] [CrossRef]
- Zuo, J.B.; An, T.Q.; Fiscella, A. A critical Kirchhoff-type problem driven by a p(·)-fractional Laplace operator with variable s(·)-order. Math. Meth. Appl. Sci. 2022, 44, 1071–1085. [Google Scholar] [CrossRef]
- Fan, Z.A. On fractional Choquard-Kirchhoff equations with subcritical or critical nonlinearities. Complex Var. Elliptic Equ. 2023, 68, 445–460. [Google Scholar] [CrossRef]
- Chen, W.J. Critical fractional p-Kirchhoff type problem with a generalized Choquard nonlinearity. J. Math. Phys. 2018, 59, 121502. [Google Scholar] [CrossRef]
- Diening, L.; Harjulehto, P.; Hästö, P.; Ružička, M. Lebesgue and Sobolev Spaces with Variable Exponents; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Willem, M. Minimax Theorems; Birkhäuser: Boston, MA, USA, 1996. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Li, S.; An, T.; Wu, Y.; Zhang, Z. A Class of Subcritical and Critical Schrödinger–Kirchhoff Equations with Variable Exponents. Fractal Fract. 2025, 9, 136. https://doi.org/10.3390/fractalfract9030136
Li S, An T, Wu Y, Zhang Z. A Class of Subcritical and Critical Schrödinger–Kirchhoff Equations with Variable Exponents. Fractal and Fractional. 2025; 9(3):136. https://doi.org/10.3390/fractalfract9030136
Chicago/Turabian StyleLi, Shuai, Tianqing An, Yue Wu, and Zhenfeng Zhang. 2025. "A Class of Subcritical and Critical Schrödinger–Kirchhoff Equations with Variable Exponents" Fractal and Fractional 9, no. 3: 136. https://doi.org/10.3390/fractalfract9030136
APA StyleLi, S., An, T., Wu, Y., & Zhang, Z. (2025). A Class of Subcritical and Critical Schrödinger–Kirchhoff Equations with Variable Exponents. Fractal and Fractional, 9(3), 136. https://doi.org/10.3390/fractalfract9030136