数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (6): 1679-1699.doi: 10.1007/s10473-020-0605-8

• 论文 • 上一篇    下一篇

RADIALLY SYMMETRIC SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING NONHOMOGENEOUS OPERATORS IN AN ORLICZ-SOBOLEV SPACE SETTING

Jae-Myoung KIM1, Yun-Ho KIM2, Jongrak LEE3   

  1. 1. Department of Mathematics Education, Andong National University, Andong 36729, Republic of Korea;
    2. Department of Mathematics Education, Sangmyung University, Seoul 03016, Republic of Korea;
    3. Department of Mathematics, Jeju National University, Jeju 63243, Republic of Korea
  • 收稿日期:2019-09-10 修回日期:2020-07-27 出版日期:2020-12-25 发布日期:2020-12-30
  • 通讯作者: Yun-Ho KIM,E-mail:kyh1213@smu.ac.kr E-mail:kyh1213@smu.ac.kr
  • 作者简介:Jae-Myoung KIM,E-mail:jmkim02@andong.ac.kr;Jongrak LEE,E-mail:jrlee@jejunu.ac.kr
  • 基金资助:
    Jae-Myoung Kim's work was supported by a Research Grant of Andong National University. Yun-Ho Kim's work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1F1A1057775). Jongrak Lee's work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07048620).

RADIALLY SYMMETRIC SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING NONHOMOGENEOUS OPERATORS IN AN ORLICZ-SOBOLEV SPACE SETTING

Jae-Myoung KIM1, Yun-Ho KIM2, Jongrak LEE3   

  1. 1. Department of Mathematics Education, Andong National University, Andong 36729, Republic of Korea;
    2. Department of Mathematics Education, Sangmyung University, Seoul 03016, Republic of Korea;
    3. Department of Mathematics, Jeju National University, Jeju 63243, Republic of Korea
  • Received:2019-09-10 Revised:2020-07-27 Online:2020-12-25 Published:2020-12-30
  • Contact: Yun-Ho KIM,E-mail:kyh1213@smu.ac.kr E-mail:kyh1213@smu.ac.kr
  • Supported by:
    Jae-Myoung Kim's work was supported by a Research Grant of Andong National University. Yun-Ho Kim's work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1F1A1057775). Jongrak Lee's work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07048620).

摘要: We investigate the following elliptic equations: $$ \begin{cases} -M\Bigl(\int_{\mathbb{R}^N}\phi(|\nabla u|^2){\rm d}x\Bigr)\text{div}(\phi^{\prime}(|\nabla u|^2)\nabla u) +|u|^{\alpha-2}u=\lambda h(x,u), \\[2mm] u(x) \rightarrow 0, \quad \text{as} \ |x| \rightarrow \infty, \end{cases} \quad \text{ in } \ \ \mathbb{R}^N, $$ where $N \geq 2$, $1 < p < q < N$, $\alpha < q$, $1 < \alpha \leq p^*q^{\prime}/p^{\prime}$ with $p^*=\frac{Np}{N-p}$, $\phi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, and $p^{\prime}$ and $q^{\prime}$ are the conjugate exponents of $p$ and $q$, respectively. We study the existence of nontrivial radially symmetric solutions for the problem above by applying the mountain pass theorem and the fountain theorem. Moreover, taking into account the dual fountain theorem, we show that the problem admits a sequence of small-energy, radially symmetric solutions.

关键词: radial solution, quasilinear elliptic equations, variational methods, Orlicz-Sobolev spaces

Abstract: We investigate the following elliptic equations: $$ \begin{cases} -M\Bigl(\int_{\mathbb{R}^N}\phi(|\nabla u|^2){\rm d}x\Bigr)\text{div}(\phi^{\prime}(|\nabla u|^2)\nabla u) +|u|^{\alpha-2}u=\lambda h(x,u), \\[2mm] u(x) \rightarrow 0, \quad \text{as} \ |x| \rightarrow \infty, \end{cases} \quad \text{ in } \ \ \mathbb{R}^N, $$ where $N \geq 2$, $1 < p < q < N$, $\alpha < q$, $1 < \alpha \leq p^*q^{\prime}/p^{\prime}$ with $p^*=\frac{Np}{N-p}$, $\phi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, and $p^{\prime}$ and $q^{\prime}$ are the conjugate exponents of $p$ and $q$, respectively. We study the existence of nontrivial radially symmetric solutions for the problem above by applying the mountain pass theorem and the fountain theorem. Moreover, taking into account the dual fountain theorem, we show that the problem admits a sequence of small-energy, radially symmetric solutions.

Key words: radial solution, quasilinear elliptic equations, variational methods, Orlicz-Sobolev spaces

中图分类号: 

  • 35J50