R^(2)上带多重狄拉克测度的非线性亥姆霍茨方程  

Nonlinear Helmholtz Equation Involving Multiple Dirac Masses in R^(2)

作　　者：马勇 陈虎元[2] Ma Yong;Chen Huyuan(Department of Computer Information Engineering,Jiangxi Normal University,Nanchang 330022;Department of Mathematics and Statistics,Jiangxi Normal University,Nanchang 330022)

机构地区：[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

出　　处：《数学物理学报（A辑）》2021年第3期652-665,共14页Acta Mathematica Scientia

基　　金：国家自然科学基金(12071189);江西省自然科学基金(20202BAB201005,20202ACBL201001);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)。

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−Δu−u=Q|u|^(p−2)u+∑_(i=1)^(N)k_(i)δ_(A_(i))(0.1)的弱解,其中p>1,k_(i)∈R\{0},i=1,…,N,Q:R^(2)→[0,+∞)是Hölder连续函数,δ_(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|α(α≤0)控制的退化,p>max{2,3(2+α)},那么存在k^(∗)>0,使得当k=∑_(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+α)}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|^(−12)控制的衰减性.Our purpose of this paper is to study weak solutions of nonlinear Helmholtz equation−Δu−u=Q|u|^(p−2)u+∑_(i=1)^(N)k_(i)δ_(A_(i))(0.1)where p>1,k_(i)∈R\{0}with i=1,…,N,Q:R^(2)→[0,+∞)is a Hölder continuous function andδ_(A_(i))is the Dirac mass concentrated at A_(i).We obtain two solutions of(0.1)if k=∑_(i=1)^(N)|k_(i)|<k^(∗)for some k^(∗)>0 when Q decays as|x|αat infinity withα≤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|^(−12)at infinity when p>max{2,4(2+α)}.

关 键 词：亥姆霍茨方程 孤立奇点 山路引理 

分 类 号：O177[理学—数学]