Existence of positive ground state solutions for Choquard equation with variable exponent growth

Gui-Dong Li; Chun-Lei Tang

doi:10.3934/dcdss.2019131

Article Contents

2019, Volume 12, Issue 7: 2035-2050. Doi: 10.3934/dcdss.2019131

This issue Previous Article Selective Pyragas control of Hamiltonian systems Next Article Solutions of nonlinear periodic Dirac equations with periodic potentials

Existence of positive ground state solutions for Choquard equation with variable exponent growth

Gui-Dong Li and
Chun-Lei Tang^,

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Chun-Lei Tang (tangcl@swu.edu.cn)
* Corresponding author: Chun-Lei Tang (tangcl@swu.edu.cn)

Received: October 31, 2017

Revised: April 30, 2018

Published: June 2019

Supported by National Natural Science Foundation of China (No.11471267).

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Abstract

In this paper, we investigate the following Choquard equation

$ \begin{equation*} -\Delta u = (I_\alpha*|u|^{f(x)})|u|^{f(x)-2}u \ \ \ \ {\rm in} \ \mathbb{R}^N, \end{equation*} $

where $ N\geq 3 $, $ \alpha\in (0,N) $ and $ I_\alpha $ is the Riesz potential. If

$ \begin{equation*} f(x) = \begin{cases} p, &x\in\Omega,\\ ( N+\alpha)/(N-2), &x\in\mathbb{R}^N \backslash\Omega, \end{cases} \end{equation*} $

where $ 1< p <\frac{N+\alpha}{N-2} $ and $ \Omega\subset\mathbb{R}^N $ is a bounded set with nonempty, we obtain the existence of positive ground state solutions by using the Nehari manifold.

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Mathematics Subject Classification: Primary: 35A15, 35B50; Secondary: 35J61.

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References

[1]	E. Acerbi and G. Mingione, Regularity results for stationary electrorheological fuids, Arch. Ration. Mech. Anal., 164 (2002), 213-259. doi: 10.1007/s00205-002-0208-7.
[2]	C. O. Alves, Existence of solution for a degenerate $p(x)$-Laplacian equation in $\mathbb{R}^N$, J. Math. Anal. Appl., 345 (2008), 731-742. doi: 10.1016/j.jmaa.2008.04.060.
[3]	C. O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrodinger equation in $\mathbb{R}^2$, J. Differential Equations, 261 (2016), 1933-1972. doi: 10.1016/j.jde.2016.04.021.
[4]	C. O. Alves, F. Gao, M. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988. doi: 10.1016/j.jde.2017.05.009.
[5]	C. O. Alves, A. Nobrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differential Equations, 55 (2016), Art. 48, 28 pp. doi: 10.1007/s00526-016-0984-9.
[6]	C. O. Alves and M. Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasilinear Choquard equation via the penalization method, Proc. Roy. Soc. Edinb. A Math., 146 (2015), 23-58. doi: 10.1017/S0308210515000311.
[7]	S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Anal., 65 (2006), 728-761. doi: 10.1016/j.na.2005.09.035.
[8]	S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), 19-36. doi: 10.1007/s11565-006-0002-9.
[9]	T. Aubin, Problémes isopérimétriques et espaces de Sobolev, J. Diferential Geom., 11 (1976), 573-598. doi: 10.4310/jdg/1214433725.
[10]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.
[11]	L. Calotă, On some quasilinear elliptic equations with critical Sobolev exponents and non-standard growth conditions, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 249-256.
[12]	A. Chambolle and P. L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), 167-188. doi: 10.1007/s002110050258.
[13]	F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math., 20 (2018), 1750037, 22 pp. doi: 10.1142/S0219199717500377.
[14]	F. Gao and M. Yang, On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents, J. Math. Anal. Appl., 448 (2017), 1006-1041. doi: 10.1016/j.jmaa.2016.11.015.
[15]	F. Gao and M. Yang, On the Brézis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math., 61 (2018), 1219-1242. doi: 10.1007/s11425-016-9067-5.
[16]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977.
[17]	G.-D. Li and C.-L. Tang, Existence of ground state solutions for Choquard equation involving the general upper critical, Commun. Pure Appl. Anal. 18 (2019) 285-300. doi: 10.3934/cpaa.2019015.
[18]	G.-D. Li and C.-L. Tang, Existence of a ground state solution for Choquard equation with the upper critical exponent, Comput. Math. Appl. 76 (2018), 2635-2647.
[19]	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.
[20]	E. H. Lieb and M. Loss, Analysis. Second Edition. Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.
[21]	P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4.
[22]	J. Liu, J.-F. Liao and C.-L. Tang, Ground state solutions for semilinear elliptic equations with zero mass in $\mathbb{R}^N$, Electron. J. Differential Equations, 2015 (2015), 1-11.
[23]	D. Lü, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal., 99 (2014), 35-48. doi: 10.1016/j.na.2013.12.022.
[24]	D. Lü, Existence and Concentration of Solutions for a Nonlinear Choquard Equation, Mediterr. J. Math., 12 (2015), 839-850. doi: 10.1007/s00009-014-0428-8.
[25]	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.
[26]	R. A. Mashiyev, B. Cekic, M. Avci and Z. Yucedag, Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition, Complex Var. Elliptic Equ., 57 (2012), 579-595. doi: 10.1080/17476933.2011.598928.
[27]	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.
[28]	I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Topology of the Universe Conference (Cleveland, OH, 1997). Classical Quantum Gravity, 15 (1998), 2733-2742. doi: 10.1088/0264-9381/15/9/019.
[29]	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.
[30]	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.
[31]	V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184. doi: 10.1016/j.jfa.2013.04.007.
[32]	V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12 pp. doi: 10.1142/S0219199715500054.
[33]	V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations, 52 (2015), 199-235. doi: 10.1007/s00526-014-0709-x.
[34]	S. Pekar, Untersuchungen über die Elektronentheorie der Kristalle, Akademie Verlag. Berlin. 1954.
[35]	P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257 (2014), 1529-1566. doi: 10.1016/j.jde.2014.05.023.
[36]	M. Råužička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics. 1748. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.
[37]	M. Riesz, L'intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math., 81 (1949), 1-223. doi: 10.1007/BF02395016.
[38]	D. Ruiz and J. Van Schaftingen, Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differential Equations, 264 (2018), 1231-1262. doi: 10.1016/j.jde.2017.09.034.
[39]	G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.
[40]	J. Van Schaftingen and J. Xia, Choquard equations under confining external potentials, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 1, 24 pp. doi: 10.1007/s00030-016-0424-8.
[41]	T. Wang and T. S. Yi, Uniqueness of positive solutions of the Choquard type equations, Appl. Anal., 96 (2017), 409-417. doi: 10.1080/00036811.2016.1138473.
[42]	M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.
[43]	H. Zhang, J. Xu and F. Zhang, Bound and ground states for a concave-convex generalized Choquard equation, Acta Appl. Math. 147 (2017), 81-93. doi: 10.1007/s10440-016-0069-y.
[44]	H. Zhang, J. Xu and F. Zhang, Existence and multiplicity of solutions for a generalized Choquard equation, Comput. Math. Appl., 73 (2017), 1803-1814. doi: 10.1016/j.camwa.2017.02.026.
[45]	V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710,877. doi: 10.1070/IM1987v029n01ABEH000958.