$\mathbb{R}^2$上带多重狄拉克测度的非线性亥姆霍茨方程

数学物理学报 ›› 2021, Vol. 41 ›› Issue (3): 652-665.

$\mathbb{R}^2$上带多重狄拉克测度的非线性亥姆霍茨方程

马勇¹(),陈虎元^2,*()

¹ 江西师范大学计算机信息工程学院南昌 330022
² 江西师范大学数学与统计学院南昌 330022

收稿日期:2020-07-26 出版日期:2021-06-01 发布日期:2021-06-09
通讯作者: 陈虎元 E-mail:mayong2020@yeah.net;chenhuyuan@yeah.net
作者简介:马勇, E-mail: mayong2020@yeah.net
基金资助:
国家自然科学基金(12071189);江西省自然科学基金(20202BAB201005);江西省自然科学基金(20202ACBL201001);江西省教育厅科学技术项目(GJJ200307);江西省重点研发计划项目(20181ACE50029)

Nonlinear Helmholtz Equation Involving Multiple Dirac Masses in $\mathbb{R}^2$

Yong Ma¹(),Huyuan Chen^2,*()

¹ Department of Computer Information Engineering, Jiangxi Normal University, Nanchang 330022
² Department of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022

Received:2020-07-26 Online:2021-06-01 Published:2021-06-09
Contact: Huyuan Chen E-mail:mayong2020@yeah.net;chenhuyuan@yeah.net
Supported by:
the NSFC(12071189);the NSF of Jiangxi Province(20202BAB201005);the NSF of Jiangxi Province(20202ACBL201001);the Science and Technology Research Project of Jiangxi Provincial Department of Education(GJJ200307);the Key R&D Plan of Jiangxi Province(20181ACE50029)

摘要/Abstract

摘要：

该文的目的是研究在二维全空间上的非线性亥姆霍茨方程 (0.1) $ -\Delta u-u=Q|u{|^{p-2}}u+\sum\limits_{i=1}^N{{k_i}}{\delta_{{A_i}}} $ 的弱解，其中$p>1$，$k_i\in$$\mathbb{R}$\{0}，i=1，…，N，Q：$\mathbb{R}^2$→[0，+∞）是Hölder连续函数，${\delta_{{A_i}}}$是集中在${{A_i}}$上的狄拉克测度.假定$Q$在无穷远处有由$|x|^{\alpha}$（$\alpha\leq 0$）控制的退化，$p>\max\{2，3（2+\alpha）\}$，那么存在$k^*>0$，使得当$k=\sum\limits_{i=1}^N{|{k_i}}| < {k^*}$时，方程（0.1）有两个弱解.它们是变号实值解，且在${{A_i}}$处具有全向性奇性.此外这里的奇性解是（0.1）对应积分方程的能量泛函的临界点，该临界点是通过山路引理得到的.最后当$p>\max\{2，4（2+\alpha）\}$时，该文使用迭代的方法获得了方程（0.1）的弱解在无穷远处有由$|x|^{-\frac12}$控制的衰减性.

关键词: 亥姆霍茨方程, 孤立奇点, 山路引理

Abstract:

Our purpose of this paper is to study weak solutions of nonlinear Helmholtz equation (0.1) $-\Delta u-u=Q|u{|^{p-2}}u+\sum\limits_{i=1}^N{{k_i}}{\delta_{{A_i}}}$ where $p>1$, $k_i\in$$\mathbb{R}$\{0} with i=1, …, N, Q: $\mathbb{R}^2$→[0, +∞) is a Hölder continuous function and ${\delta_{{A_i}}}$ is the Dirac mass concentrated at ${{A_i}}$.We obtain two solutions of (0.1) if $k=\sum\limits_{i=1}^N{|{k_i}}| < {k^*}$ for some $k^*$>0 when $Q$ decays as $|x|^{α}$ at infinity with $α ≤ $ 0 and $p$>max{2, 3(2+$α$)}. These two sequences of solutions of (0.1) are sign-changing real-valued solutions with isotropic singularity at ${{A_i}}$ by applying Mountain Pass Theorem to an related integral equation. By using the iteration technique, we obtain the decays of solution of (0.1) controlled by $|x|^{-\frac12}$ at infinity when $p>\max\{2, 4(2+\alpha)\}$.

Key words: Helmholtz equation, Isolated singularity, Mountain pass theorem

中图分类号:

O177

马勇,陈虎元. $\mathbb{R}^2$上带多重狄拉克测度的非线性亥姆霍茨方程[J]. 数学物理学报, 2021, 41(3): 652-665.

Yong Ma,Huyuan Chen. Nonlinear Helmholtz Equation Involving Multiple Dirac Masses in $\mathbb{R}^2$[J]. Acta mathematica scientia,Series A, 2021, 41(3): 652-665.

参考文献 32

1	Alsholm P , Schmidt G . Spectral and scattering theory for Schrödinger operators. Archive Ration Mech Anal, 1971, 40, 281- 311 doi: 10.1007/BF00252679
2	Ao W , Chan H , Gonzélez M , Wei J . Existence of positive weak solutions for fractional Lane-Emden equations with prescribed singular sets. Calc Var PDE, 2018, 57 (6): 1- 25 doi: 10.1007/s00526-018-1425-8
3	Bénilan Ph , Brezis H . Nonlinear problems related to the Thomas-Fermi equation. J Evolution Eq, 2003, 3, 673- 770 doi: 10.1007/s00028-003-0117-8
4	Bidaut-Véron M , Pohozaev S . Nonexistence results and estimates for some nonlinear elliptic problems. J Anal Math, 2001, 84, 1- 49 doi: 10.1007/BF02788105
5	Brezis H, Lions P L. A Note on Isolated Singularities for Linear Elliptic Equations//Brezis H. Mathematical Analysis and Applications. New York: Acad Press, 1981: 263-266
6	Brezis H , Véron L . Removable singularities of some nonlinear elliptic equations. Arch Rational Mech Anal, 1980, 75, 1- 6 doi: 10.1007/BF00284616
7	Caffarelli L , Gidas B , Spruck J . Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth. Comm Pure Appl Math, 1989, 42, 271- 297 doi: 10.1002/cpa.3160420304
8	Chen C , Lin C . Existence of positive weak solutions with a prescribed singular set of semi-linear elliptic equations. J Geom Anal, 1999, 9 (2): 221- 246 doi: 10.1007/BF02921937
9	Chen H , Huang X , Zhou F . Fast decaying solutions of Lane-Emden equations involving nonhomogeneous potential. Adv Nonlin Stud, 2020, doi: 10.1515/ans-2020-2071
10	Chen H , Quaas A . Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results. J London Math Soc, 2018, 97 (2): 196- 221 doi: 10.1112/jlms.12104
11	Chen H, Evéquoz G, Weth T. Complex solutions and stationary scattering for the nonlinear Helmholtz equation. 2019, ArXiv: 1911.09557
12	Chen H , Zhou F . Classification of isolated singularities of positive solutions for Choquard equations. J Diff Eq, 2016, 261, 6668- 6698 doi: 10.1016/j.jde.2016.08.047
13	Enciso A , Peralta-Salas D . Bounded solutions to the Allen-Cahn equation with level sets of any compact topology. Analysis and PDE, 2016, 9 (6): 1433- 1446 doi: 10.2140/apde.2016.9.1433
14	Evéquoz G , Weth T . Real solutions to the nonlinear Helmholtz equation with local nonlinearity. Arch Rational Meth Anal, 2014, 211, 359- 388 doi: 10.1007/s00205-013-0664-2
15	Evéquoz G , Weth T . Dual variational methods and nonvanishing for the nonlinear Helmholtz equation. Adv Math, 2015, 280, 690- 728 doi: 10.1016/j.aim.2015.04.017
16	Evéquoz G, Yesil T. Dual ground state solutions for the critical nonlinear Helmholtz equation. 2017, ArXiv: 1707.00959
17	Gidas B , Spruck J . Global and local behaviour of positive solutions of nonlinear elliptic equations. Comm Pure Appl Math, 1981, 34, 525- 598 doi: 10.1002/cpa.3160340406
18	Guitiérrez S . Non trivial L^q solutions to the Ginzburg-Landau equation. Math Ann, 2004, 328, 1- 25 doi: 10.1007/s00208-003-0444-7
19	Lions P . Isolated singularities in semilinear problems. J Diff Eq, 1980, 38 (3): 441- 450 doi: 10.1016/0022-0396(80)90018-2
20	Lieb E H , Simon B . The Thomas-Fermi theory of atoms, molecules and solids. Adv Math, 1977, 23, 22- 116 doi: 10.1016/0001-8708(77)90108-6
21	Mandel R . The limiting absorption principle for periodic differential operators and applications to nonlinear Helmholtz equations. Comm Math Phys, 2019, 368 (2): 799- 842 doi: 10.1007/s00220-019-03363-1
22	Mandel R , Montefusco E , Pellacci B . Oscillating solutions for nonlinear Helmholtz equations. Z Angew Math Phys, 2017, 68, Article number: 121 doi: 10.1007/s00033-017-0859-8
23	Mazzeo R , Pacard F . A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis. J Diff Geom, 1996, 44 (2): 331- 370
24	Mousomi B , Nguyen P . On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures. Adv Nonlinear Anal, 2020, 9 (1): 1480- 1503 doi: 10.1515/anona-2020-0060
25	Ponce A C , Wilmet N . Schrödinger operators involving singular potentials and measure data. J Diff Eq, 2017, 263, 3581- 3610 doi: 10.1016/j.jde.2017.04.039
26	Stein E M . Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton University Press, 1970
27	Mandel R , Scheider D . Dual variational methods for a nonlinear Helmholtz system. Nonli Differ Equa Appl NoDEA, 2018, 25, Article number: 13 doi: 10.1007/s00030-018-0504-z
28	Wang Y . Isolated singularities of solutions of defocusing Hartree equation. Nonlinear Anal TMA, 2017, 156, 70- 81 doi: 10.1016/j.na.2017.01.019
29	Vazquez J L . On a semilinear equation in $\mathbb{R}^2$ involving bounded measures. Proc Roy Soc Edinburgh, 1983, 95, 181- 202 doi: 10.1017/S0308210500012907
30	Véron L . Singular solutions of some nonlinear elliptic equations. Nonlinear Anal TMA, 1981, 5, 225- 242 doi: 10.1016/0362-546X(81)90028-6
31	Véron L. Elliptic Equations Involving Measures, Stationary Partial Differential Equations. Vol Ⅰ. Amsterdam: Elsevier, 2008: 593-712
32	Zemach C , Odeh F . Uniqueness of radiative solutions to the Schrödinger wave equation. Arch Rational Mech Anal, 1960, 5, 226- 237 doi: 10.1007/BF00252905

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