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

Jinlong Bai; Desheng Li; Chunqiu Li

doi:10.3934/cpaa.2019151

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2019, Volume 18, Issue 6: 3351-3365. Doi: 10.3934/cpaa.2019151

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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
^* Corresponding author

Received: November 2018

Revised: February 2019

Published: May 2019

This work is supported by NSF of China 11471240, 11871368.

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Abstract

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} $.

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Mathematics Subject Classification: 34C23, 35B32, 35K55, 37B30.

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