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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

Jinlong Bai 1, , Desheng Li 1, and  Chunqiu Li 2,,

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} $
.
Keywords: Conley index theory, semilinear elliptic equations, multiple solutions, resonance.
Mathematics Subject Classification: 34C23, 35B32, 35K55, 37B30.
Citation: Jinlong Bai, Desheng Li, Chunqiu Li. A note on multiplicity of solutions near resonance of semilinear elliptic equations. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3351-3365. doi: 10.3934/cpaa.2019151
References:
[1]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977. doi: 10.1007/BFb0087685.

[2]

X. Chang and Y. Li, Existence and multiplicity of nontrivial solutions for semilinear elliptic Dirichlet problems across resonance, Topol. Methods Nonlinear Anal., 36 (2010), 285-310. 

[3]

R. Chiappinelli, J. Mawhin and R. Nugari, Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities, Nonlinear Anal. TMA, 18 (1992), 1099-1112.  doi: 10.1016/0362-546X(92)90155-8.

[4]

C. Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics 38, Amer. Math. Soc., Providence RI, 1978. doi: 10.1090/cbms/038.

[5]

C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc., 158 (1971), 35-61.  doi: 10.2307/1995770.

[6]

F. de Paiva and E. Massa, Semilinear elliptic problems near resonance with a nonprincipal eigenvalue, J. Math. Anal. Appl., 342 (2008), 638-650.  doi: 10.1016/j.jmaa.2007.12.053.

[7]

M. Filippakis, L. Gasiński and N. Papageorgiou, A multiplicity result for semilinear resonant elliptic problems with nonsmooth potential, Nonlinear Anal. TMA, 61 (2005), 61-75.  doi: 10.1016/j.na.2004.11.012.

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math. 840, Springer Verlag, Berlin, 1981. doi: 10.1007/BFb0089647.

[9]

C. Li, D. Li and Z. Zhang, Dynamic bifurcation from infinity of nonlinear evolution equations, SIAM J. Appl. Dyn. Syst., 16 (2017), 1831-1868.  doi: 10.1137/16M1107358.

[10]

D. Li, G. Shi and X. Song, Linking theorems of local semiflows on complete metric spaces, arXiv: 1312.1868.

[11]

D. Li and Z. Wang, Local and global dynamic bifurcations of nonlinear evolution equations, Indiana Univ. Math. J., 67 (2018), 583-621.  doi: 10.1512/iumj.2018.67.7292.

[12]

T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science-A: Monographs and Treatises, vol. 53, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.

[13]

T. Ma and S. Wang, Dynamic bifurcation of nonlinear evolution equations, Chinese Ann. Math. Ser. B, 26 (2005), 185-206.  doi: 10.1142/S0252959905000166.

[14]

E. Massa and R. Rossato, Multiple solutions for an elliptic system near resonance, J. Math. Anal. Appl., 420 (2014), 1228-1250.  doi: 10.1016/j.jmaa.2014.06.043.

[15]

J. Mawhin and K. Schmitt, Landesman-Lazer type problems at an eigenvalue of odd multiplicity, Results Math., 14 (1988), 138-146.  doi: 10.1007/BF03323221.

[16]

C. McCord, Poincaré-Lefschetz duality for the homolopy Conley index, Trans. Amer. Math. Soc., 329 (1992), 233-252.  doi: 10.2307/2154086.

[17]

K. Mischaikow and M. Mrozek, Conley Index, Handbook of Dynamical Systems, vol. 2, Elsevier, New York, 2002,393-460. doi: 10.1016/S1874-575X(02)80030-3.

[18]

M. Mrozek and R. Srzednicki, On time-duality of the Conley index, Results Math., 24 (1993), 161-167.  doi: 10.1007/BF03322325.

[19]

N. Papageorgiou and F. Papalini, Multiple solutions for nearly resonant nonlinear Dirichlet problems, Potential Anal., 37 (2012), 247-279.  doi: 10.1007/s11118-011-9255-8.

[20]

K. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-72833-4.

[21]

J. Saut and R. Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations, 4 (1979), 293-319.  doi: 10.1080/03605307908820096.

[22]

K. Schmitt and Z. Wang, On bifurcation from infinity for potential operators, Differential and Integral Equations, 4 (1991), 933-943. 

[23]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[24]

J. Su and C. Tang, Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues, Nonlinear Anal. TMA, 44 (2001), 311-321.  doi: 10.1016/S0362-546X(99)00265-5.

show all references

References:
[1]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977. doi: 10.1007/BFb0087685.

[2]

X. Chang and Y. Li, Existence and multiplicity of nontrivial solutions for semilinear elliptic Dirichlet problems across resonance, Topol. Methods Nonlinear Anal., 36 (2010), 285-310. 

[3]

R. Chiappinelli, J. Mawhin and R. Nugari, Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities, Nonlinear Anal. TMA, 18 (1992), 1099-1112.  doi: 10.1016/0362-546X(92)90155-8.

[4]

C. Conley, Isolated Invariant Sets and the Morse Index, Regional Conference Series in Mathematics 38, Amer. Math. Soc., Providence RI, 1978. doi: 10.1090/cbms/038.

[5]

C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc., 158 (1971), 35-61.  doi: 10.2307/1995770.

[6]

F. de Paiva and E. Massa, Semilinear elliptic problems near resonance with a nonprincipal eigenvalue, J. Math. Anal. Appl., 342 (2008), 638-650.  doi: 10.1016/j.jmaa.2007.12.053.

[7]

M. Filippakis, L. Gasiński and N. Papageorgiou, A multiplicity result for semilinear resonant elliptic problems with nonsmooth potential, Nonlinear Anal. TMA, 61 (2005), 61-75.  doi: 10.1016/j.na.2004.11.012.

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math. 840, Springer Verlag, Berlin, 1981. doi: 10.1007/BFb0089647.

[9]

C. Li, D. Li and Z. Zhang, Dynamic bifurcation from infinity of nonlinear evolution equations, SIAM J. Appl. Dyn. Syst., 16 (2017), 1831-1868.  doi: 10.1137/16M1107358.

[10]

D. Li, G. Shi and X. Song, Linking theorems of local semiflows on complete metric spaces, arXiv: 1312.1868.

[11]

D. Li and Z. Wang, Local and global dynamic bifurcations of nonlinear evolution equations, Indiana Univ. Math. J., 67 (2018), 583-621.  doi: 10.1512/iumj.2018.67.7292.

[12]

T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science-A: Monographs and Treatises, vol. 53, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.

[13]

T. Ma and S. Wang, Dynamic bifurcation of nonlinear evolution equations, Chinese Ann. Math. Ser. B, 26 (2005), 185-206.  doi: 10.1142/S0252959905000166.

[14]

E. Massa and R. Rossato, Multiple solutions for an elliptic system near resonance, J. Math. Anal. Appl., 420 (2014), 1228-1250.  doi: 10.1016/j.jmaa.2014.06.043.

[15]

J. Mawhin and K. Schmitt, Landesman-Lazer type problems at an eigenvalue of odd multiplicity, Results Math., 14 (1988), 138-146.  doi: 10.1007/BF03323221.

[16]

C. McCord, Poincaré-Lefschetz duality for the homolopy Conley index, Trans. Amer. Math. Soc., 329 (1992), 233-252.  doi: 10.2307/2154086.

[17]

K. Mischaikow and M. Mrozek, Conley Index, Handbook of Dynamical Systems, vol. 2, Elsevier, New York, 2002,393-460. doi: 10.1016/S1874-575X(02)80030-3.

[18]

M. Mrozek and R. Srzednicki, On time-duality of the Conley index, Results Math., 24 (1993), 161-167.  doi: 10.1007/BF03322325.

[19]

N. Papageorgiou and F. Papalini, Multiple solutions for nearly resonant nonlinear Dirichlet problems, Potential Anal., 37 (2012), 247-279.  doi: 10.1007/s11118-011-9255-8.

[20]

K. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-72833-4.

[21]

J. Saut and R. Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations, 4 (1979), 293-319.  doi: 10.1080/03605307908820096.

[22]

K. Schmitt and Z. Wang, On bifurcation from infinity for potential operators, Differential and Integral Equations, 4 (1991), 933-943. 

[23]

G. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[24]

J. Su and C. Tang, Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues, Nonlinear Anal. TMA, 44 (2001), 311-321.  doi: 10.1016/S0362-546X(99)00265-5.

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