Zhi-Yun Tang, Zeng-Qi Ou. INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1912-1917. doi: 10.11948/20190286
Citation: |
Zhi-Yun Tang, Zeng-Qi Ou. INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1912-1917. doi: 10.11948/20190286
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INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM
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1.
School of Mathematics and Statistics, Southwest University, Tiansheng Road, 400710, Beibei, Chongqing, China
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Corresponding authors:
Email address: yvonne50@qq.com(Z. Tang); Email address: ouzengq707@sina.com (Z. Ou)
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Abstract
Consider a class of nonlocal problems
$ \left\{\begin{array}{lr}
-\left(a-b \int_{\Omega}|\nabla u|^{2} d x\right) \Delta u=f(x, u), & x \in \Omega, \\
u=0, & 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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