In this study, we consider the following quasilinear Choquard equation with singularity
1 Introduction and main results
In this study, we investigate the following quasilinear Choquard equation with singularity
In recent years, the study on the quasilinear Schrödinger equation (1.2) is always a topic of great interest. Mathematicians have established several methods to treat equation (1.3), for example, the dual approach, the perturbation method, and the Nehari method, see for instance [14,15, 16,17,18, 19,20,21, 22,23,24, 25,26,27, 28,29], and the references therein. However, the system (1.1) with Choquard type nonlinearity has only been studied in [30,31].
It is remarkable that there are few papers investigating quasilinear equation with singularity. To the best of our knowledge, it only appears in , J. Marcos do Ó and A. Moameni established the singular quasilinear Schrödinger equation
To the best of our knowledge, there seems to be little progress on the existence of a positive solution for quasilinear Choquard equation with singularity. By the motivation of the above work, in our study, we establish the existence of a positive solution for problem (1.1) with singularity. First, the nonlinearity of problem (1.1) is nonlocal, and it is much more difficult to obtain the existence of positive solutions. Second, we investigate the relationships between quasilinear Choquard equation involving and without convolution, which makes our studies more interesting. At last, we obtain the asymptotic behavior of solutions as .
Before stating our main result, we suppose that the functions and satisfy the following assumptions:
satisfies , where is a constant.
for all .
is a nonnegative function.
Suppose that , , and hold, then equation (1.1) admits a unique solution in .
The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented, and in Section 3, we give the proof of our main results.
2 Variational setting and preliminaries
To prove our conclusion, we give some basic notations and preliminaries. First, we can rewrite (1.1) as
If we make the change of variable , we may rewrite equation in the form
(See ) The function f satisfies the following properties:
is uniquely defined function and invertible;
and for all ;
for all ;
for all ;
the function is strictly convex;
there exists a positive constant such that
for each , we have for all ;
the function is strictly decreasing for and ;
the function is strictly increasing for and .
3 Proof of Theorem 1.1
To prove Theorem 1.1, we need the following results.
Suppose that are satisfied, then (1.1) has the global minimizer in . In other words, there exists such that .
By the Sobolev inequality, Hölder inequality and Lemma 2.1 yield
Proof of Theorem 1.1
We divide the proof into three parts.
We will prove that for any(3.5)since and is nonnegative.
Dividing (3.5) by and passing to the liminf as , then we can get from Fatou’s Lemma that(3.6)Since(3.7)
(3.8)For any and , set and . Then, using ( 3.7) and ( 3.8) with lead to(3.9)Analogous to the proof of [ 32, Theorem 1], we obtain . Since , the strong maximum principle implies , and is a solution of problem ( 1.1).
Proof of Theorem 1.2
From the proof of Lemma 3.1 and Theorem 1.1, we can get that is allowed. Therefore, under the conditional assumptions of Theorem 1.2, equation (1.1) has a unique positive solution , i.e., for any , we obtain
The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.
Funding information: This work was partially supported by the Fundamental Research Funds for the National Natural Science Foundation of China 11671403 and Guizhou University of Finance and Economics of 2019XYB15.
Conflict of interest: Authors state no conflict of interest.
 S. Kurihura , Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Jpn. 50 (1981), no. 10, 3262–3267, https://doi.org/10.1143/JPSJ.50.3262 . Search in Google Scholar
 E. Laedke , K. Spatschek , and L. Stenflo , Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys. 24 (1983), no. 12, 2764–2769, https://doi.org/10.1063/1.525675 . Search in Google Scholar
 H. Lange , M. Poppenberg , and H. Teismann , Nash-Moser methods for the solution of quasilinear Schrödinger equations, Commun. Partial Differ. Equ. 24 (1999), no. 7–8, 1399–1418, https://doi.org/10.1080/03605309908821469 . Search in Google Scholar
 A. Borovskii and A. Galkin , Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys. 77 (1993), 562–573. Search in Google Scholar
 E. Gloss , Existence and concentration of positive solutions for a quasilinear equation in RN , J. Math. Anal. Appl. 371 (2010), no. 2, 465–484, https://doi.org/10.1016/j.jmaa.2010.05.033 . Search in Google Scholar
 A. de Bouard , N. Hayashi , and J. Saut , Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys. 189 (1997), 73–105, https://doi.org/10.1007/s002200050191 . Search in Google Scholar
 A. Litvak and A. Sergeev , One-dimensional collapse of plasma waves, JETP Lett. 27 (1978), no. 10, 517–520. Search in Google Scholar
 A. Nakamura , Damping and modification of exciton solitary waves, J. Phys. Soc. Jpn. 42 (1977), no. 6, 1824–1835, https://doi.org/10.1143/JPSJ.42.1824 . Search in Google Scholar
 F. G. Bass and N. N. Nasanov , Nonlinear electromagnetic spin waves, Phys. Rep. 189 (1990), no. 4, 165–223, https://doi.org/10.1016/0370-1573(90)90093-H . Search in Google Scholar
 R. Hasse , A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Physik B. 37 (1980), 83–87, https://doi.org/10.1007/BF01325508 . Search in Google Scholar
 V. G. Makhankov and V. K. Fedyanin , Non-linear effects in quasi-one-dimensional models of condensed matter theory, Physics Reports 104 (1984), no. 1, 1–86, https://doi.org/10.1016/0370-1573(84)90106-6 . Search in Google Scholar
 B. Ritchie , Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E 50 (1994), no. 2, R687–R689, https://doi.org/10.1103/PhysRevE.50.R687 . Search in Google Scholar
 H. Lange , B. Toomire , and P. Zweifel , Time-dependent dissipation in nonlinear Schrödinger systems, J. Math. Phys. 36 (1995), no. 3, 1274–1283, https://doi.org/10.1063/1.531120 . Search in Google Scholar
 C. O. Alves and M. Yang , Multiplicity and concentration of solutions for a quasilinear Choquarde quation, J. Math. Phys. 55 (2014), no. 6, 061502, https://doi.org/10.1063/1.4884301 . Search in Google Scholar
 E. Silva and G. Vieira , Quasilenear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differ. Equ. 39 (2010), 1–33, https://doi.org/10.1007/s00526-009-0299-1 . Search in Google Scholar
 E. Silva and G. Vieira , Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal. 72 (2010), no. 6, 2935–2949, https://doi.org/10.1016/j.na.2009.11.037 . Search in Google Scholar
 V. Moroz and J. Van Schaftingen , Groundstates of nonlinear Choquard equations Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), no. 5, 1550005, https://doi.org/10.1142/S0219199715500054 . Search in Google Scholar
 M. Millem , Minimax Theorems, Birkhöuser, Berlin, 1996. Search in Google Scholar
 V. Moroz and J. Van Schaftingen , Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), 6557–6579, https://doi.org/10.1090/S0002-9947-2014-06289-2 . Search in Google Scholar
 V. Moroz and J. Van Schaftingen , Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), no. 2, 153–184, https://doi.org/10.1016/j.jfa.2013.04.007 . Search in Google Scholar
 D. Cao and S. Peng , A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms, J. Diff. Eqns. 193 (2003), no. 2, 424–434, https://doi.org/10.1016/S0022-0396(03)00118-9 . Search in Google Scholar
 S. Cingolani , M. Clapp , and S. Secchi , Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys. 63 (2012), 233–248, https://doi.org/10.1007/s00033-011-0166-8 . Search in Google Scholar
 M. Poppenberg , K. Schmitt , and Z. Wang , On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ. 14 (2002), 329–344, https://doi.org/10.1007/s005260100105 . Search in Google Scholar
 H. Brözis and T. Kato , Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. 9 (1979), 137–151. Search in Google Scholar
 N. Hirano , C. Saccon , and N. Shioji , Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Diff. Eqns. 245 (2008), no. 8, 1997–2037, https://doi.org/10.1016/j.jde.2008.06.020 . Search in Google Scholar
 D. Lü , Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations, Commun. Pure. Appl. Anal. 15 (2016), no. 5, 1781–1795, https://doi.org/10.3934/cpaa.2016014 . Search in Google Scholar
 Z. Shen , F. Gao , and M. Yang , Multiple solutions for nonhomogeneous Choquard equation involving Hardy-Littlewood-Sobolev critical exponent, Z. Angew. Math. Phys. 68 (2017), 61, https://doi.org/10.1007/s00033-017-0806-8 . Search in Google Scholar
 M. Colin and L. Jeanjean , Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal. 56 (2004), no. 2, 213–226, https://doi.org/10.1016/j.na.2003.09.008 . Search in Google Scholar
 S. Chen and X. Wu , Existence of positive solutions for a class of quasilinear Schrödinger equations of Choquard type, J. Math. Anal. Appl. 475 (2019), no. 2, 1754–1777, https://doi.org/10.1016/j.jmaa.2019.03.051 . Search in Google Scholar
 X. Yang , W. Zhang , and F. Zhao , Existence and muliplicity of solutions for a quasilinear Choquard equation via perturbation method, J. Math. Phys. 59 (2018), no. 8, 081503, https://doi.org/10.1063/1.5038762 . Search in Google Scholar
 C. Alves and M. Yang , Multiplicity and concentration behavior of solutions for a quasilinear Choquard equation via penalization method, Proc. Roy. Soc. Edinburgh Sect. A. 146 (2016), no. 6, 23–58. Search in Google Scholar
 J. Marcos do Ó and A. Moameni , Solutions for singular quasilinear Schrödinger quation with one parameter, Commun. Pure. Appl. Anal. 9 (2010), no. 4, 1011–1023, https://doi.org/10.3934/cpaa.2010.9.1011 . Search in Google Scholar
 J. Chen , B. Cheng , and X. Huang , Ground state solutions for a class of quasilinear Schrödinger equations with Choquard type nonlinearity, Appl. Math. Lett. 102 (2020), 106141, https://doi.org/10.1016/j.aml.2019.106141 . Search in Google Scholar
 J. Liu , Y. Wang , and Z. Wang , Soliton solutions for quasilinear Schrödinger equations: II, J. Diff. Eqns. 187 (2003), no. 2, 473–493, https://doi.org/10.1016/S0022-0396(02)00064-5 . Search in Google Scholar
 X. Yang , W. Wang , and F. Zhao , Infinitely many radial and non-radial solutions to a quasilinear Schrödinger equation, Nonlinear Anal. 114 (2015), no. 11, 158–168, https://doi.org/10.1016/j.na.2014.11.015 . Search in Google Scholar
 X. Li , S. Ma , and G. Zhang , Existence and qualitative properties of solutions for Choquard equations with a local term, Nonlinear Anal. Real World Appl. 45 (2019), 1–25, https://doi.org/10.1016/j.nonrwa.2018.06.007 . Search in Google Scholar
© 2021 Liuyang Shao and Yingmin Wang, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.