分数阶Choquard方程正解的存在性、多重性和集中现象

摘要/Abstract

摘要：

该文考虑了下面的次临界的分数阶Choquard方程 $ \begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-N}(|x|^{-\mu}\ast F(u))f(u), \quad x\in\mathbb{R} ^{N} \end{eqnarray*} $ 正解的存在性、多重性和集中现象, 这里$\varepsilon>0$是一个常数, $s\in (0, 1)$, $(-\Delta)^{s}$是分数阶Laplace算子, 位势$V:\mathbb{R} ^{N}\rightarrow\mathbb{R} $是正的且有全局极小, $0<\mu<\min\{4s, N\}$, 非线性项$f\in C^{1}(\mathbb{R} , \mathbb{R} )$是次临界增长的, $F$是$f$的原函数.该文的主要研究方法是变分法和Ljusternik-Schnirelmann理论.

关键词: 分数阶Choquard方程, 变分法, Ljusternik-Schnirelmann理论, 正解, 集中现象

Abstract:

We are concerned with the existence, multiplicity and concentration of positive solutions for the following fractional Choquard equation with subcritical nonlinearity $\begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-N}(|x|^{-\mu}\ast F(u))f(u), \quad x\in \mathbb{R} ^{N}, \end{eqnarray*} $ where $\varepsilon>0$ is a parameter, $s\in(0, 1)$, $(-\Delta)^{s}$ is the fractional Laplace operator, $V:\mathbb{R} ^{N}\rightarrow\mathbb{R} $ is a positive potential having global minimum, $0<\mu<\min\{4s, N\}$, and $F$ is the primitive of $f\in C^{1}(\mathbb{R} , \mathbb{R} )$ which is subcritical growth. The main research methods of this article are variational method and the Ljusternik-Schnirelmann theory.

Key words: Fractional Choquard equation, Variational method, Ljusternik-Schnirelmann theory, Positive solution, Concentrating phenomenon.

中图分类号:

O175.2

张伟强,赵培浩. 分数阶Choquard方程正解的存在性、多重性和集中现象[J]. 数学物理学报, 2022, 42(2): 470-490.

Weiqiang Zhang,Peihao Zhao. Existence, Multiplicity and Concentration of Positive Solutions for a Fractional Choquard Equation[J]. Acta mathematica scientia,Series A, 2022, 42(2): 470-490.

参考文献 34

1	Alves C O , Gao F S , Squassina M , Yang M B . Singularly perturbed critical Choquard equations. J Differential Equations, 2017, 263 (7): 3943- 3988 doi: 10.1016/j.jde.2017.05.009
2	Alves C O, Miyagaki O H. Existence and concentration of solution for a class of fractional elliptic equation in $ {{\Bbb R}} ^{N} $ via penalization method. Calc Var Partial Differential Equations, 2016, 55(3), Article number: 47
3	Alves C O , Ambrosio V . A multiplicity result for a nonlinear fractional Schödinger equation in $ {{\Bbb R}} ^{N} $ without the Ambrosetti-Rabinowitz condition. J Math Anal Appl, 2018, 466 (1): 498- 522 doi: 10.1016/j.jmaa.2018.06.005
4	Ambrosio V . Multiplicity and concentration results for a fractional Choquard equation via penalization method. Potential Anal, 2019, 50 (1): 55- 82 doi: 10.1007/s11118-017-9673-3
5	Ambrosio V . Concentration phenomena for a fractional Choquard equation with magnetic field. Dyn Partial Differ Equa, 2019, 16 (2): 125- 149 doi: 10.4310/DPDE.2019.v16.n2.a2
6	Applebaum D . L$ \acute{\rm e} $vy processes-from probability to finance and quantum groups. Not Amer Math Soc, 2004, 51 (11): 1336- 1347
7	Benci V , Cerami G . Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. Calc Var Partial Differential Equations, 1994, 2 (1): 29- 48 doi: 10.1007/BF01234314
8	Cassani D , Zhang J J . Choquard-type equations with Hardy-Littlewood-Sobolev upper-critical growth. Adv Nonlinear Anal, 2019, 8 (1): 1184- 1212
9	Cingolani S , Lazzo M . Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J Differential Equations, 2000, 160 (1): 118- 138 doi: 10.1006/jdeq.1999.3662
10	Chen S T , Tang X H , Wei J Y . Nehari-type ground state solutions for a Choquard equation with doubly critical exponents. Adv Nonlinear Anal, 2021, 10 (1): 152- 171
11	Di Nezza E , Palatucci G , Valdinoci E . Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136 (5): 521- 573 doi: 10.1016/j.bulsci.2011.12.004
12	Felmer F , Quaas A , Tan J G . Positive solutions of the nonlinear Schrödinger equation with the fractional Laplace. Proc Roy Soc Edinburgh Sect A, 2012, 142 (6): 1237- 1262 doi: 10.1017/S0308210511000746
13	Figueiredo G M, Siciliano G. A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for fractional Schödinger equation in $ {{\Bbb R}} ^{N} $. NoDEA Nonlinear Differential Equations Appl, 2016, 23(2), Article number:12
14	Gao F S , Yang M B , Zhou J H . Existence of multiple semiclassical solutions for a critical Choquard equation with indefinite potential. Nonlinear Anal, 2020, 195: 111817 doi: 10.1016/j.na.2020.111817
15	He X M , Radulescu V D . Small linear perturbations of fractional Choquard equations with critical exponent. J Differential Equations, 2021, 282: 481- 540 doi: 10.1016/j.jde.2021.02.017
16	Lan F Q , He X M . The Nehari manifold for a fractional critical Choquard equation involving sign-changing weight functions. Nonlinear Anal, 2019, 180: 236- 263 doi: 10.1016/j.na.2018.10.010
17	Laskin N . Fractional Schrödinger equation. Phys Rev E, 2002, 66 (5): 056108 doi: 10.1103/PhysRevE.66.056108
18	Lieb E, Loss M. Analysis. Providence, RI: American Mathematical Society, 2001
19	Moroz V , Van Schaftingen J . Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J Funct Anal, 2013, 265 (2): 153- 184 doi: 10.1016/j.jfa.2013.04.007
20	Moroz V , Van Schaftingen J . Existence of groundstates for a class of nonlinear Choquard equations. Trans Amer Math Soc, 2015, 367 (9): 6557- 6579
21	Moroz V , Van Schaftingen J . Semi-classical states for the Choquard equation. Calc Var Partial Differential Equations, 2015, 52 (1/2): 199- 235
22	Moser J . A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations. Comm Pure Appl Math, 1960, 13: 457- 468 doi: 10.1002/cpa.3160130308
23	Palatucci G , Pisante A . Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc Var Partial Differential Equations, 2014, 50 (3/4): 799- 829
24	Rabinowitz P H . On a class of nonlinear Schrödinger equations. Z Angew Math Phys, 1992, 43 (2): 270- 291 doi: 10.1007/BF00946631
25	Su Y , Wang L , Chen H B , Liu S L . Multiplicity and concentration results for fractional Choquard equations: doubly critical case. Nonlinear Anal, 2020, 198: 111872 doi: 10.1016/j.na.2020.111872
26	Silvestre L . Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm Pure Appl Math, 2007, 60 (1): 67- 112 doi: 10.1002/cpa.20153
27	Tang X H , Cheng B T . Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J Differential Equations, 2016, 261 (4): 2384- 2402 doi: 10.1016/j.jde.2016.04.032
28	Tang X H , Chen S T . Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions. Adv Nonlinear Anal, 2020, 9 (1): 413- 437
29	Willem M . Minimax Theorems. Boston: Birkhäuser, 1996
30	Xiang M Q , Radulescu V D , Zhang B L . A critical fractional Choquard-Kirchhoff problem with magnetic field. Commun Contemp Math, 2019, 21 (4): 1850004 doi: 10.1142/S0219199718500049
31	Yang Z P , Zhao F K . Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth. Adv Nonlinear Anal, 2021, 10 (1): 732- 774
32	Zhang H , Zhang F B . Multiplicity and concentration of solutions for Choquard equations with critical growth. J Math Anal Appl, 2020, 481 (1): 123457 doi: 10.1016/j.jmaa.2019.123457
33	Zhang H , Wang J , Zhang F B . Semiclassical states for fractional Choquard equations with critical growth. Commun Pure Appl Anal, 2019, 18 (1): 519- 538 doi: 10.3934/cpaa.2019026
34	Zhang Y P , Tang X H , Radulescu V D . High perturbations of Choquard equations with critical reaction and variable growth. Proc Amer Math Soc, 2021, 149 (9): 3819- 3835 doi: 10.1090/proc/15469

[1]	胡蝶,高琦. 含有对数非线性项Kirchhoff方程多解的存在性[J]. 数学物理学报, 2022, 42(2): 401-417.
[2]	夏晨阳,王振辉,程志波. 一类带阻尼的吸引型奇性Duffing方程周期正解的存在性[J]. 数学物理学报, 2022, 42(1): 131-138.
[3]	储昌木,蒙璐. 一类带有变指数增长的半线性椭圆方程正解的存在性[J]. 数学物理学报, 2021, 41(6): 1779-1790.
[4]	张丽红,杨泽栋,王国涛,BaleanuDumitru. 一类k-Hessian方程解的存在性和渐近稳定性[J]. 数学物理学报, 2021, 41(5): 1357-1371.
[5]	杨先勇,唐先华,顾光泽. 带有临界增长或超临界增长的分数阶Choquard方程解的存在性和多重性[J]. 数学物理学报, 2021, 41(3): 702-722.
[6]	贾小尧,娄振洛. Hénon型椭圆系统多个非径向对称解的存在性[J]. 数学物理学报, 2021, 41(3): 723-728.
[7]	徐家发,罗洪林,刘立山. 一类具有p-Laplacian算子的分数阶差分方程边值问题的正解[J]. 数学物理学报, 2021, 41(2): 402-414.
[8]	任晶,翟成波. 基于变分法的回火分数阶脉冲微分系统分析[J]. 数学物理学报, 2021, 41(2): 415-426.
[9]	刘春根,钟余友. 有限图上高阶Yamabe型方程的非平凡解[J]. 数学物理学报, 2021, 41(1): 39-45.
[10]	邓义华. ${\mathbb R}$^$N$上一类Kirchhoff型方程径向对称正解的存在性[J]. 数学物理学报, 2021, 41(1): 142-148.
[11]	姚旺进. 基于变分方法的脉冲微分方程耦合系统解的存在性和多重性[J]. 数学物理学报, 2020, 40(3): 705-716.
[12]	徐家发. 具有半正非线性项的分数阶差分方程组边值问题的正解[J]. 数学物理学报, 2020, 40(1): 132-145.
[13]	李容星,王文清,曾小雨. 带椭球势阱的Kirchhoff型方程的变分问题[J]. 数学物理学报, 2019, 39(6): 1323-1333.
[14]	吉蕾,廖家锋. 一类带临界指数的非齐次Kirchhoff型问题第二个正解的存在性[J]. 数学物理学报, 2019, 39(5): 1094-1101.
[15]	张晶. 一类次临界Bose-Einstein凝聚型方程组的渐近收敛行为和相位分离[J]. 数学物理学报, 2019, 39(3): 441-450.