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Volume 10, Number 5, 2020, Pages 1912-1917                                                                DOI:10.11948/20190286
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.
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