Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (3): 652-665.

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

Yong Ma1(),Huyuan Chen2,*()   

  1. 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;
  • 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)


Our purpose of this paper is to study weak solutions of nonlinear Helmholtz equation $-\Delta u-u=Q|u{|^{p-2}}u+\sum\limits_{i=1}^N{{k_i}}{\delta_{{A_i}}}$ 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

CLC Number: 

  • O177