Acta mathematica scientia,Series B ›› 2020, Vol. 40 ›› Issue (6): 1679-1699.

• Articles •

### 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).

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.

CLC Number:

• 35J50
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