Infinitely Many Small Energy Solutions to Schrödinger-Kirchhoff Type Problems Involving the Fractional -Laplacian in
Abstract
:1. Introduction
- (K1)
- satisfies , where is a constant.
- (K2)
- There exists such that for any .
- (f1)
- as uniformly for all .
- (f2)
- uniformly for almost all .
- (f3)
- There exists a constant such that
- (f4)
- There is a positive function such that
2. Preliminaries
- (1)
- If , then ;
- (2)
- If , then .
- (1)
- if is a bounded Lipschitz domain and such that in
- (2)
- for any uniformly continuous function with in and ;
- (3)
- for any satisfying in
- (P)
- , , and .
- (1)
- there is a compact embedding ;
- (2)
- for any measurable function with in , there is a compact embedding if .
3. Existence of Solutions
- (A1)
- and for all .
- (A2)
- satisfying that .
- (B1)
- satisfies the Carathéodory condition and there exist a positive constant and a nonnegative function such that
- (B2)
- There are , such that
- (B3)
- There exist , , with for all and a positive function such that
- (D1)
- ;
- (D2)
- ;
- (D3)
- as ;
- (D4)
- fulfills the -condition for every ,
4. Conclusions
- (L1)
- , where ;
- (L2)
- there exists a positive constant such that for almost all and , where ;
- (L3)
- for all .
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Boureanu, M.-M.; Preda, F. Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions. Nonlinear Differ. Equ. Appl. 2012, 19, 235–251. [Google Scholar] [CrossRef] [Green Version]
- Diening, L.; Harjulehto, P.; Hästö, P.; Růžička, M. Lebesgue and Sobolev Spaces with Variable Exponents. In Lecture Notes in Mathematics; Springer: Berlin, Germany, 2011; Volume 2017. [Google Scholar]
- Kim, I.H.; Kim, Y.-H.; Park, K. Existence and multiplicity of solutions for Schrödinger-Kirchhoff type problems involving the fractional p(·)-Laplacian in . Bound. Value Probl. 2020, 2020, 1–24. [Google Scholar] [CrossRef]
- Mihăilescu, M.; Rxaxdulescu, V. A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc. R. Soc. Ser. A 2006, 462, 2625–2641. [Google Scholar] [CrossRef] [Green Version]
- Růžička, M. Electrorheological Fluids: Modeling and Mathematical Theory. In Lecture Notes in Mathematics; Springer: Berlin, Germany, 2000; Volume 1748. [Google Scholar]
- Caffarelli, L. Non-local equations, drifts and games. Nonlinear Partial Differ. Equ. Abel Symp. 2012, 7, 37–52. [Google Scholar]
- Gilboa, G.; Osher, S. Nonlocal operators with applications to image processing. Multiscale Model. Simul. 2008, 7, 1005–1028. [Google Scholar] [CrossRef] [Green Version]
- Laskin, N. Fractional quantum mechanics and Levy path integrals. Phys. Lett. A 2000, 268, 298–305. [Google Scholar] [CrossRef] [Green Version]
- Di Nezza, E.; Palatucci, G.; Valdinoci, E. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 2012, 136, 521–573. [Google Scholar] [CrossRef]
- Servadei, R.; Valdinoci, E. Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 2012, 389, 887–898. [Google Scholar] [CrossRef] [Green Version]
- Azroul, E.; Benkirane, A.; Boumazourh, A.; Shimi, M. Existence results for fractional p(x,·)-Laplacian problem via the Nehari manifold approach. Appl. Math. Optim. 2021, 84, 1527–1547. [Google Scholar] [CrossRef]
- Azroul, E.; Benkirane, A.; Shimi, M. Existence and multiplicity of solutions for fractional p(x,·)-Kirchhoff-type problems in . Appl. Anal. 2021, 100, 2029–2048. [Google Scholar] [CrossRef]
- Bahrouni, A. Comparison and sub-supersolution principles for the fractional p(x)-Laplacian. J. Math. Anal. Appl. 2018, 458, 1363–1372. [Google Scholar] [CrossRef]
- Bahrouni, A.; Rădulescu, V. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discret. Contin. Dyn. Syst. Ser. S 2018, 11, 379–389. [Google Scholar] [CrossRef] [Green Version]
- Biswas, R.; Bahrouni, A.; Fiscella, A. Fractional double phase Robin problem involving variable-order exponents and logarithm-type nonlinearity. Math. Methods Appl. Sci. 2022, 45, 11272–11296. [Google Scholar] [CrossRef]
- Biswas, R.; Tiwari, S. Variable order nonlocal Choquard problem with variable exponents. Complex Var. Elliptic Equ. 2021, 66, 853–875. [Google Scholar] [CrossRef]
- Bonaldo, L.M.M.; Hurtado, E.J.; Miyagaki, O.H. Multiplicity results for elliptic problems involving nonlocal integrodifferential operators without Ambrosetti-Rabinowitz condition. Discret. Contin. Dyn. Syst. 2022, 42, 3329–3353. [Google Scholar] [CrossRef]
- Cheng, Y.; Ge, B.; Agarwal, R.P. Variable-order fractional Sobolev spaces and nonlinear elliptic equations with variable exponents. J. Math. Phys. 2020, 61, 071507. [Google Scholar] [CrossRef]
- Chung, N.T. Eigenvalue problems for fractional p(x,y)-Laplacian equations with indefinite weight. Taiwan. J. Math. 2019, 23, 1153–1173. [Google Scholar] [CrossRef]
- Ho, K.; Kim, Y.-H. A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional p(·)-Laplacian. Nonlinear Anal. 2019, 188, 179–201. [Google Scholar] [CrossRef]
- Ho, K.; Kim, Y.-H. The concentration-compactness principles for Ws,p(·,·)() and application. Adv. Nonlinear Anal. 2021, 10, 816–848. [Google Scholar] [CrossRef]
- Kaufmann, U.; Rossi, J.D.; Vidal, R. Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians. Electron. J. Qual. Theory Differ. Equ. 2017, 76, 1–10. [Google Scholar] [CrossRef]
- Lee, J.; Kim, J.-M.; Kim, Y.-H.; Scapellato, A. On multiple solutions to a non-local Fractional p(·)-Laplacian problem with concave-convex nonlinearities. Adv. Cont. Discr. Mod. 2022, 2022, 1–25. [Google Scholar] [CrossRef]
- Zuo, J.; Yang, L.; Liang, S. A variable-order fractional p(·)-Kirchhoff type problem in . Math. Methods Appl. Sci. 2020, 44, 3872–3889. [Google Scholar]
- Liang, S.; Pucci, P.; Zhang, B. Existence and multiplicity of solutions for critical nonlocal equations with variable exponents. Appl. Anal. 2022. [Google Scholar] [CrossRef]
- Zuo, J.; An, T.; Fiscella, A. A critical Kirchhoff-type problem driven by a p(·)-fractional Laplace operator with variable s(·)-order. Math. Methods Appl. Sci. 2020, 44, 1071–1085. [Google Scholar] [CrossRef]
- Kirchhoff, G. Vorlesungen über Mechanik; Teubner: Leipzig, Germany, 1883. [Google Scholar]
- Colasuonno, F.; Pucci, P. Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations. Nonlinear Anal. 2011, 74, 5962–5974. [Google Scholar] [CrossRef]
- Dai, G.W.; Hao, R.F. Existence of solutions of a p(x)-Kirchhoff-type equation. J. Math. Anal. Appl. 2009, 359, 275–284. [Google Scholar] [CrossRef] [Green Version]
- Liu, D.C. On a p(x)-Kirchhoff-type equation via fountain theorem and dual fountain theorem. Nonlinear Anal. 2010, 72, 302–308. [Google Scholar] [CrossRef]
- Pucci, P.; Xiang, M.; Zhang, B. Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in . Calc. Var. Partial Differ. Equ. 2015, 54, 2785–2806. [Google Scholar] [CrossRef]
- Xiang, M.; Zhang, B.; Ferrara, M. Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian. J. Math. Anal. Appl. 2015, 424, 1021–1041. [Google Scholar] [CrossRef]
- Alves, C.O.; Liu, S.B. On superlinear p(x)-Laplacian equations in . Nonlinear Anal. 2010, 73, 2566–2579. [Google Scholar] [CrossRef]
- Liu, S.B. On ground states of superlinear p-Laplacian equations in . J. Math. Anal. Appl. 2010, 61, 48–58. [Google Scholar] [CrossRef] [Green Version]
- Liu, S.B.; Li, S.J. Infinitely many solutions for a superlinear elliptic equation. Acta Math. Sin. (Chin. Ser.) 2003, 46, 625–630. (In Chinese) [Google Scholar]
- Tan, Z.; Fang, F. On superlinear p(x)-Laplacian problems without Ambrosetti and Rabinowitz condition. Nonlinear Anal. 2012, 75, 3902–3915. [Google Scholar] [CrossRef]
- Lin, X.; Tang, X.H. Existence of infinitely many solutions for p-Laplacian equations in . Nonlinear Anal. 2013, 92, 72–81. [Google Scholar] [CrossRef]
- Hurtado, E.J.; Miyagaki, O.H.; Rodrigues, R.S. Existence and multiplicity of solutions for a class of elliptic equations without Ambrosetti-Rabinowitz type conditions. J. Dyn. Diff. Equ. 2018, 30, 405–432. [Google Scholar] [CrossRef]
- Kim, Y.-H. Multiple solutions to Kirchhoff-Schrödinger equations involving the p(·)-Laplace type operator. AIMS Math. 2023, in press. [Google Scholar] [CrossRef]
- Jeanjean, L. On the existence of bounded Palais-Smale sequences and application to a Landsman-Lazer type problem set on . Proc. R. Soc. Edinb. Sect. A Math. 1999, 129, 787–809. [Google Scholar] [CrossRef] [Green Version]
- Miyagaki, O.H.; Souto, M.A.S. Superlinear problems without Ambrosetti and Rabinowitz growth condition. J. Differ. Equ. 2008, 245, 3628–3638. [Google Scholar] [CrossRef] [Green Version]
- Biswas, R.; Bahrouni, S.; Carvalho, M.L. Fractional double phase Robin problem involving variable order-exponents without Ambrosetti–Rabinowitz condition. Z. Angew. Math. Phys. 2022, 73, 1–25. [Google Scholar] [CrossRef]
- Biswas, R.; Tiwari, S. On a class of Kirchhoff-Choquard equations involving variable-order fractional p(·)-Laplacian and without Ambrosetti-Rabinowitz type condition. Topol. Methods Nonlinear Anal. 2021, 58, 403–439. [Google Scholar] [CrossRef]
- Cen, J.; Kim, S.J.; Kim, Y.-H.; Zeng, S. Multiplicity results of solutions to the double phase anisotropic variational problems involving variable exponent. Adv. Differ. Equ. 2023, in press. [Google Scholar]
- Stegliński, R. Notes on aplications of the dual fountain theorem to local and nonlocal elliptic equations with variable exponent. Opusc. Math. 2022, 42, 751–761. [Google Scholar] [CrossRef]
- Teng, K. Multiple solutions for a class of fractional Schrödinger equations in . Nonlinear Anal. Real World Appl. 2015, 21, 76–86. [Google Scholar] [CrossRef]
- Willem, M. Minimax Theorems; Birkhauser: Basel, Switzerland, 1996. [Google Scholar]
- Lee, J.I.; Kim, J.-M.; Kim, Y.-H.; Lee, J. Multiplicity of weak solutions to non-local elliptic equations involving the fractional p(x)-Laplacian. J. Math. Phys. 2020, 61, 011505. [Google Scholar] [CrossRef]
- Fan, X.; Zhao, D. On the spaces Lp(x)(Ω) and Wm,p(x)(Ω). J. Math. Anal. Appl. 2001, 263, 424–446. [Google Scholar] [CrossRef] [Green Version]
- Kováčik, O.; Rxaxkosník, J. On spaces Lp(x) and Wk,p(x). Czechoslovak Math. J. 1991, 41, 592–618. [Google Scholar] [CrossRef]
- Edmunds, D.E.; Rákosník, J. Sobolev embedding with variable exponent. Stud. Math. 2000, 143, 267–293. [Google Scholar] [CrossRef]
- Kim, J.-M.; Kim, Y.-H. Multiple solutions to the double phase problems involving concave–convex nonlinearities. AIMS Math. 2023, 8, 5060–5079. [Google Scholar] [CrossRef]
- Fabian, M.; Habala, P.; Hajék, P.; Montesinos, V.; Zizler, V. Banach Space Theory: The Basis for Linear and Nonlinear Analysis; Springer: New York, NY, USA, 2011. [Google Scholar]
- Zhou, Y.; Wang, J.; Zhang, L. Basic Theory of Fractional Differential Equations, 2nd ed.; World Scientific Publishing Co. Pte. Ltd.: Singapore, 2017. [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. |
© 2023 by the author. 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
Kim, Y.-H.
Infinitely Many Small Energy Solutions to Schrödinger-Kirchhoff Type Problems Involving the Fractional
Kim Y-H.
Infinitely Many Small Energy Solutions to Schrödinger-Kirchhoff Type Problems Involving the Fractional
Kim, Yun-Ho.
2023. "Infinitely Many Small Energy Solutions to Schrödinger-Kirchhoff Type Problems Involving the Fractional
Kim, Y. -H.
(2023). Infinitely Many Small Energy Solutions to Schrödinger-Kirchhoff Type Problems Involving the Fractional