Abstract
This article deals with the fractional Choquard equation involving exponential growth as follows
where \(\varepsilon \) a positive parameter, \(s\in (0,1)\), \(\mu \in (0, N)\) and \((-\Delta )_{p}^{s}\) is a fractional p-Laplace operator with \(p=\frac{N}{s}\ge 2.\) The function h is only continuous and has exponential growth. In addition, the potential function Z satisfies some appropriate conditions. We use the Trudinger–Moser inequality to deal with the function h involving exponential growth. Together with the Ljusternik–Schnirelmann category theory and variational method, the multiplicity and concentration behavior of positive solutions are obtained for the above problem. As far as we know, our results seem to be new for the fractional \(\frac{N}{s}\)-Laplacian Choquard equation.
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References
Alves, C., Cassani, D., Tarsi, C., Yang, M.: Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in \(\mathbb{R} ^2\). J. Differ. Equ. 261, 1933–1972 (2016)
Alves, C.: Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method. Proc. R. Soc. Edinb. A 146, 23–58 (2016)
Alves, C., Gao, F., Squassina, M., Yang, M.: Singularly perturbed critical Choquard equations. J. Differ. Equ. 263, 3943–3988 (2017)
Ambrosio, V., Isernia, T.: Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional \(p\)-Laplace. Discrete Contin. Dyn. Syst. 38, 5835–5881 (2018)
Ambrosio, V.: Multiplicity and concentration results for a fractional Choquard equation via penalization method. Potential Anal. 50, 55–82 (2019)
Ambrosio, V.: On the multiplicity and concentration of positive solutions for a \(p\)-fractional Choquard equation in \(\mathbb{R} ^N\). Comput. Appl. Math. 78, 2593–2617 (2019)
Ambrosio, V.: Nonlinear fractional Schrödinger equations in \(\mathbb{R}^{N}.\) In: Frontiers in Elliptic and Parabolic Problems. Birkhäuser/Springer, Cham (2021)
Bisci, G., Thin, N.V., Vilasi, L.: On a class of nonlocal Schrödinger equations with exponential growth. Adv. Differ. Equ. 27, 571–610 (2022)
Caffarelli, L., Silvesytre, L.: An extension problems related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
Chen, S., Li, L., Yang, Z.: Multiplicity and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent. RACSAM 114, 33 (2020)
Chen, S., Shu, M., Tang, X., Wen, L.: Planar Schrödinger–Poisson system with critical exponential growth in the zero mass case. J. Differ. Equ. 327, 448–480 (2022)
Cingolani, S., Tanaka, K.: Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well. Rev. Mat. Iberoam. 35, 1885–1924 (2019)
Clemente, R., Albuquerque, J.C.D., Barboza, E.: Existence of solutions for a fractional Choquard-type equation in \(\mathbb{R} ^N\) with critical exponential growth. Z. Angew. Math. Phys. 72, 16 (2021)
de Böer, E., Miyagaki, O.H.: Existence and multiplicity of solutions for the fractional \(p\)-Laplacian Choquard logarithmic equation involving a nonlinearity with exponential critical and subcritical growth. J. Math. Phys. 62, 051507 (2021)
Del Pezzo, L., Quaas, A.: A Hopf’s lemma and a strong minimum principle for the fractional \(p\)-Laplacian. J. Differ. Equ. 263, 765–778 (2017)
Figueiredo, G., Bisci, G.M., Servadei, R.: The effect of the domain topology on the number of solutions of fractional Laplace problems. Calc. Var. Partial Differ. Equ. 57, 1–24 (2018)
Floer, A., Weinstein, A.: Non spreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)
Fröhlich, H.: Theory of electrical breakdown in ionic crystal. Proc. R. Soc. A 160, 230–241 (1937)
Fröhlich, H.: Electrons in lattice fields. Adv. Phys. 3, 325–361 (1954)
Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional \(p\)-Laplacian. Rev. Mat. Iberoam. 32, 1353–1392 (2016)
Jones, K.: Newtonian quantum gravity. Aust. J. Phys. 48, 1055–1081 (1995)
Lia, Q., Yang, Z.: Multiple solutions for a class of fractional quasi-linear equations with critical exponential growth in \(\mathbb{R} ^N,\). Complex Var. Elliptic Equ. 61, 969–983 (2016)
Lieb, E.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1976/77)
Lieb, E., Loss, M.: Analysis: Graduate Studies in Mathematics. AMS, Providence (2001)
Lions, P.: The Choquard equation and related questions. Nonlinear Anal. Theory Methods Appl. 4, 1063–1072 (1980)
Liu, Z., Rădulescu, V., Tang, C., Zhang, J.: Another look at planar Schrödinger–Newton systems. J. Differ. Equ. 328, 65–104 (2022)
Meng, Y., He, X.: Multiplicity of concentrating solutions for Choquard equation with critical growth. J. Geom. Anal. 33, 78 (2023)
Moroz, V., Schaftingen, J.V.: A guide to the Choquard equation. J. Fixed Point Theory Appl. 19, 773–813 (2017)
Moroz, I., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger–Newton equations. Class. Quantum Gravity 15, 2733–2742 (1998)
Nezza, E.D., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Papageorgiou, N.S., Zhang, J., Zhang, W.: Global existence and multiplicity of solutions for nonlinear singular eigenvalue problems. Discrete Contin. Dyn. Syst. S (2024). https://doi.org/10.3934/dcdss.2024018
Papageorgiou, N.S., Rădulescu, V.D., Zhang, W.: Global existence and multiplicity for nonlinear Robin eigenvalue problems. Results Math. 78, 133 (2023)
Pekar, S.: Untersuchung über die Elektronentheorie der Kristalle, p. 2. Akademie Verlag, Berlin (1954)
Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relativ. Gravit. 28, 581–600 (1996)
Pucci, P., Xiang, M., Zhang, B.: Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional \(p\)-Laplacian in \(\mathbb{R} ^N\). Calc. Var. Partial Differ. Equ. 54, 2785–2806 (2015)
Qin, D., Rădulescu, V.D., Tang, X.H.: Ground states and geometrically distinct solutions for periodic Choquard–Pekar equations. J. Differ. Equ. 275, 652–683 (2021)
Rabinowitz, P.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)
Singh, G.: Nonlocal perturbations of the fractional Choquard equation. Adv. Numer. Anal. 8, 694–706 (2017)
Su, Y., Wang, L., Chen, H., Liu, S.: Multiplicity and concentration results for fractional Choquard equations: doubly critical case. Nonlinear Anal. 198, 111872 (2020)
Szulkin, A., Weth, T.: The method of Nehari manifold. In: Gao, D.Y., Motreanu, D. (eds.) Handbook of Nonconvex Analysis and Applications, pp. 597–632. International Press, Boston (2010)
Thin, N.: Singular Trudinger–Moser inequality and fractional \(p\)-Laplace equation in \(\mathbb{R} ^N\). Nonlinear Anal. 196, 111756 (2020)
Thin, N.: Multiplicity and concentration of solutions to a fractional \(p\)-Laplace problem with exponential growth. Ann. Fenn. Math. 47, 603–639 (2022)
Willem, M.: Minimax Theorems. Birkhäuser, Basel (1996)
Ye, H.: The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in \(\mathbb{R} ^N\). J. Math. Anal. Appl. 431, 935–954 (2015)
Yang, Z., Zhao, F.: Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth. Adv. Nonlinear Anal. 10, 732–774 (2021)
Yuan, S., Tang, X., Zhang, J., Zhang, L.: Semiclassical states of fractional Choquard equations with exponential critical growth. J. Geom. Anal. 32, 290 (2022)
Yuan, S., Rădulescu, V.D., Tang, X., Zhang, L.: Concentrating solutions for singularly perturbed fractional (N/s)-Laplacian equations with nonlocal reaction. Forum Math. 36, 783–810 (2024)
Zhang, C.: Trudinger–Moser inequalities in Fractional Sobolev–Slobodeckij spaces and multiplicity of weak solutions to the Fractional-Laplacian equation. Adv. Nonlinear Stud. 19, 197–217 (2019)
Zhang, H., Zhang, F.: Multiplicity and concentration of solutions for Choquard equations with critical growth. J. Math. Anal. Appl. 481, 123457 (2020)
Zhang, J., Lü, W., Lou, Z.: Multiplicity and concentration behavior of solutions of the critical Choquard equation. Appl. Anal. 100, 167–190 (2021)
Zhang, J., Zhang, Y.: An infinite sequence of localized semiclassical states for nonlinear Maxwell–Dirac system. J. Geom. Anal. 34, 277 (2024)
Zhang, X., Sun, X., Liang, S., Thin Nguyen, V.: Existence and concentration of solutions to a Choquard equation involving fractional \(p\)-Laplacian via penalization method. J. Geom. Anal. 34, 90 (2024)
Acknowledgements
S. Shi was supported by NSFC (Grant No. 12271203) and Science and Technology Development Project of Jilin Province (Grant No. YDZJ202101ZYTS141). Thin Van Nguyen is supported by Thai Nguyen University of Education.
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Liang, S., Shi, S. & Van Nguyen, T. Multiplicity and Concentration Properties for Fractional Choquard Equations with Exponential Growth. J Geom Anal 34, 367 (2024). https://doi.org/10.1007/s12220-024-01815-2
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DOI: https://doi.org/10.1007/s12220-024-01815-2
Keywords
- Critical exponential growth
- Fractional p-Laplace
- Mountain pass theorem
- Trudinger–Moser inequality
- Variational method