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 Volume 10, Number 5, 2020, Pages 1912-1917 Infinitely many solutions for a nonlocal problem Zhiyun Tang,Zengqi Ou Keywords:Nonlocal problems, infinitely many solutions, the $(P.S.)_c$ condition, the equivariant link theorem. Abstract: Consider a class of nonlocal problems $$\left \{\begin{array}{ll} -(a-b\int_{\Omega}|\nabla u|^2dx)\Delta u= f(x,u),& \textrm{x \in\Omega},\u=0, & \textrm{x \in\partial\Omega}, \end{array} \right.$$ where $a>0, b>0$,~$\Omega\subset \mathbb{R}^N$ is a bounded open domain, $f:\overline{\Omega} \times \mathbb R \longrightarrow \mathbb R$ is a Carath$\acute{\mbox{e}}$odory function. Under suitable conditions, the equivariant link theorem without the $(P.S.)$ condition due to Willem is applied to prove that the above problem has infinitely many solutions, whose energy increasingly tends to $a^2/(4b)$, and they are neither large nor small. PDF      Download reader