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November  2019, 18(6): 3351-3365. doi: 10.3934/cpaa.2019151

A note on multiplicity of solutions near resonance of semilinear elliptic equations

 1 School of Mathematics, Tianjin University, Tianjin, 300072, China 2 Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang, 325035, China

* Corresponding author

Received  November 2018 Revised  February 2019 Published  May 2019

Fund Project: This work is supported by NSF of China 11471240, 11871368

In this paper we are concerned with the multiplicity of solutions near resonance for the following nonlinear equation:
 $-\Delta u = \lambda u+f(x,u)$
associated with the Dirichlet boundary condition, where
 $f$
satisfies some appropriate conditions. We will treat this problem in the framework of dynamical systems. It will be shown that there exist a one-sided neighborhood
 $\Lambda_-$
of the eigenvalue
 $\mu_k$
of the Laplacian operator and a dense subset
 ${\mathcal D}$
of
 $\mathbb{R}$
such that the equation has at least four distinct nontrivial solutions generically for
 $\lambda\in\Lambda_- \cap {\mathcal D}$
.
Citation: Jinlong Bai, Desheng Li, Chunqiu Li. A note on multiplicity of solutions near resonance of semilinear elliptic equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3351-3365. doi: 10.3934/cpaa.2019151
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