Existence of least energy solutions for a quasilinear Choquard equation

Vincenzo Ambrosio; Giuseppina Autuori; Teresa Isernia

doi:10.3934/cpaa.2024061

Article Contents

2024, Volume 23, Issue 11: 1661-1678. Doi: 10.3934/cpaa.2024061

This issue Previous Article Global existence for certain fourth order evolution equations Next Article The existence and concentration behavior of positive ground state solutions for a class of Choquard type equations involving local and nonlocal operators

Existence of least energy solutions for a quasilinear Choquard equation

Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, via brecce bianche 12, 60131 Ancona, Italy

^*Corresponding author: Teresa Isernia
^*Corresponding author: Teresa Isernia

Received: March 2024

Revised: July 2024

Early access: August 2024

Published: November 2024

The authors are partially supported by INdAM-GNAMPA Project CUP E53C23001670001.

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Abstract

The present paper is devoted to the quasilinear Choquard equation driven by the $ p $-Laplacian operator

$ -\Delta_{p}u + |u|^{p-2}u = (I_{\alpha} * G(u)) G'(u) \quad \mbox{ in } \mathbb{R}^{N}, $

where $ 2\leq p<N $, $ I_{\alpha} $ denotes the Riesz potential of order $ \alpha\in (0, N) $, and $ G\in \mathcal{C}^{1}( \mathbb{R}, \mathbb{R}) $. Assuming Berestycki–Lions type conditions on $ G $, we prove the existence of a least energy solution $ u\in W^{1, p}( \mathbb{R}^N) $ by means of variational methods. Moreover, we establish some qualitative properties of $ u $ when $ G $ is even and non–decreasing.

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Mathematics Subject Classification: Primary: 35A15, 35J92; Secondary: 35B06, 35B38.

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References

[1]	R. A. Adams and J. J. F. Fournier, Sobolev spaces, Pure Appl. Math. (Amst.), 140, Elsevier/Academic Press, Amsterdam, 2003.
[2]	C. O. Alves and M. Yang, Multiplicity and concentration of solutions for a quasilinear Choquard equation, J. Math. Phys., 55 (2014), 21 pp.
[3]	C. O. Alves and M. Yang, Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differ. Equ., 257 (2014), 4133-4164. doi: 10.1016/j.jde.2014.08.004.
[4]	C. O. Alves and M. Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 23-58.
[5]	V. Ambrosio, On the multiplicity and concentration of positive solutions for a $p$-fractional Choquard equation in $ \mathbb{R}^N$, Comput. Math. Appl., 78 (2019), 2593-2617.
[6]	V. Ambrosio, Regularity and Pohožaev identity for the Choquard equation involving the $p$–Laplacian operator, Appl. Math. Lett., 145 (2023), 9 pp.
[7]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[8]	L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597. doi: 10.1016/0362-546X(92)90023-8.
[9]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equation, Universitext, Springer, New York, 2011.
[10]	A. El Hamidi and J. M. Rakotoson, Compactness and quasilinear problems with critical exponents, Differ. Integral Equ., 18 (2005), 1201-1220.
[11]	L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659. doi: 10.1016/S0362-546X(96)00021-1.
[12]	L. Jeanjean and K. Tanaka, A remark on least energy solutions in $ \mathbb{R}^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1.
[13]	E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105. doi: 10.1002/sapm197757293.
[14]	E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997.
[15]	P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4.
[16]	L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.
[17]	G. P. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301. doi: 10.1017/S0308210500012191.
[18]	I. M. Moroz, R. Penrose and P. Tod, Spherically–symmetric solutions of the Schrödinger–Newton equations, Topology of the Universe Conference (Cleveland, OH, 1997), Class. Quantum Grav., 15 (1998), 2733-2742. doi: 10.1088/0264-9381/15/9/018.
[19]	V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
[20]	V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579. doi: 10.1090/S0002-9947-2014-06289-2.
[21]	V. Moroz and J. Van Schaftingen, Semi–classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52 (2015), 199-235. doi: 10.1007/s00526-014-0709-x.
[22]	V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813. doi: 10.1007/s11784-016-0373-1.
[23]	S. Pekar, Untersuchungen über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin (1954).
[24]	J. Simon, Régularité de la solution d'une équation non linéaire dans $ \mathbb{R}^{N}$, Journées d'Analyse Non Linéaire (Proc. Conf., Besançon, 1977), Lecture Notes in Math., 665, Springer, Berlin, 1978.
[25]	N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Commun. Pure Appl. Math., 20 (1967), 721-747. doi: 10.1002/cpa.3160200406.
[26]	J. Van Schaftingen, Approximation of symmetrizations and symmetry of critical points, Topol. Methods Nonlinear Anal., 28 (2006), 61-85.
[27]	M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston, MA, 1996.
[28]	M. Willem, Functional Analysis-Fundamentals and Applications, Cornerstones, Birkhäuser/Springer, New York, 2022.