Existence and multiplicity of solutions for a class of damped-like fractional differential system

Jie Xie; Xingyong Zhang; Cuiling Liu; Danyang Kang; Jie Xie; Xingyong Zhang; Cuiling Liu; Danyang Kang

doi:10.3934/math.2020272

AIMS Mathematics

2020, Volume 5, Issue 5: 4268-4284. doi: 10.3934/math.2020272

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Research article

Existence and multiplicity of solutions for a class of damped-like fractional differential system

1.
Faculty of Science, Kunming University of Science and Technology, Yunnan 650500, China
2.
School of Mathematics and Statistics, Central South University, Hunan 410083, China

Received: 10 January 2020 Accepted: 20 April 2020 Published: 07 May 2020
MSC : 34B15, 34B10

In this paper, we obtain some results about the existence and multiplicity of weak solutions for a class of damped-like fractional differential system with a parameter $\lambda$. When the nonlinear term is subquadratic only near the origin, we obtain that system has a ground state weak solution $u_\lambda$ if $\lambda$ is in some given interval, and when the nonlinear term is also even near the origin, then for each $\lambda>0$, system has infinitely many weak solutions $\{u_n^\lambda\}$ with $\|u_n^\lambda\|\to 0$ as $n\to \infty$. We mainly use Ekeland's variational principle and a variant of Clark's theorem together with a cut-off technique to prove our results.
- Damped-like fractional differential system,
- Riemann-Liouville fractional integrals,
- Ekeland’s variational principle,
- Clark’s theorem
Citation: Jie Xie, Xingyong Zhang, Cuiling Liu, Danyang Kang. Existence and multiplicity of solutions for a class of damped-like fractional differential system[J]. AIMS Mathematics, 2020, 5(5): 4268-4284. doi: 10.3934/math.2020272

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Abstract

In this paper, we obtain some results about the existence and multiplicity of weak solutions for a class of damped-like fractional differential system with a parameter $\lambda$. When the nonlinear term is subquadratic only near the origin, we obtain that system has a ground state weak solution $u_\lambda$ if $\lambda$ is in some given interval, and when the nonlinear term is also even near the origin, then for each $\lambda>0$, system has infinitely many weak solutions $\{u_n^\lambda\}$ with $\|u_n^\lambda\|\to 0$ as $n\to \infty$. We mainly use Ekeland's variational principle and a variant of Clark's theorem together with a cut-off technique to prove our results.

References

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