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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 |
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 $ |
| $ \Lambda_- $ |
| $ \mu_k $ |
| $ {\mathcal D} $ |
| $ \mathbb{R} $ |
| $ \lambda\in\Lambda_- \cap {\mathcal D} $ |
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 & 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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