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

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.