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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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##### References:
 [1] Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489 [2] Kelei Wang. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 3805-3816. doi: 10.3934/era.2021062 [3] Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387 [4] Jiabao Su, Zhaoli Liu. A bounded resonance problem for semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 431-445. doi: 10.3934/dcds.2007.19.431 [5] Nikolaos S. Papageorgiou, Calogero Vetro, Francesca Vetro. Multiple solutions for (p, 2)-equations at resonance. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 347-374. doi: 10.3934/dcdss.2019024 [6] Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629 [7] Ketty A. De Rezende, Mariana G. Villapouca. Discrete conley index theory for zero dimensional basic sets. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056 [8] Kung-Ching Chang, Zhi-Qiang Wang, Tan Zhang. On a new index theory and non semi-trivial solutions for elliptic systems. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 809-826. doi: 10.3934/dcds.2010.28.809 [9] Rong Xiao, Yuying Zhou. Multiple solutions for a class of semilinear elliptic variational inclusion problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 991-1002. doi: 10.3934/jimo.2011.7.991 [10] Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795 [11] Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations & Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081 [12] Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601 [13] Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801 [14] Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439 [15] Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886 [16] David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335 [17] Massimo Grossi. On the number of critical points of solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 4215-4228. doi: 10.3934/era.2021080 [18] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2505-2518. doi: 10.3934/cpaa.2020272 [19] Zhiguo Wang, Yiqian Wang, Daxiong Piao. A new method for the boundedness of semilinear Duffing equations at resonance. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3961-3991. doi: 10.3934/dcds.2016.36.3961 [20] M. C. Carbinatto, K. Mischaikow. Horseshoes and the Conley index spectrum - II: the theorem is sharp. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 599-616. doi: 10.3934/dcds.1999.5.599

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