# zbMATH — the first resource for mathematics

Some perturbation results of Kirchhoff type equations via Morse theory. (English) Zbl 07254121
Summary: In this paper, we consider the following Kirchhoff type equation: $\begin{cases} - (a+b \int_{\varOmega } \vert \nabla u \vert^2\,dx ) \Delta u= f(x,u)& \text{in } \varOmega , \\ u=0 & \text{on } \partial \varOmega , \end{cases}$ where $$a,b>0$$ are constants and $$\varOmega \subset \mathbb{R}^N (N=1,2,3)$$ is a bounded domain with smooth boundary $$\partial \Omega$$. By applying Morse theory, we obtain some existence and multiplicity results of nontrivial solutions for either $$a$$ or $$b$$ being sufficiently small.
##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
##### Keywords:
Kirchhoff equations; nontrivial solutions; Morse theory
Full Text:
##### References:
 [1] Brezis, H.; Nirenberg, L., $$H^1$$ versus $$C^1$$ local minimizers, C. R. Acad. Sci., Sér. 1 Math., 317, 465-472 (1993) · Zbl 0803.35029 [2] Chang, K.-C., Infinite Dimensional Morse Theory and Multiple Solution Problems (1993), Boston: Birkhäuser, Boston [3] Chang, K.-C., Methods in Nonlinear Analysis (2005), Berlin: Springer, Berlin [4] Chang, K.-C.; Ghoussoub, N., The Conley index and the critical groups via an extension of Gromoll-Meyer theory, Topol. Methods Nonlinear Anal., 7, 77-93 (1996) · Zbl 0898.58006 [5] Chang, K.-C.; Wang, Z.-Q., Notes on the bifurcation theorem, J. Fixed Point Theory Appl., 1, 195-208 (2007) · Zbl 1139.58008 [6] Chen, C.; Huang, J.; Liu, L., Multiple solutions to the nonhomogeneous p-Kirchhoff elliptic equation with concave-convex nonlinearities, Appl. Math. Lett., 26, 754-759 (2013) · Zbl 1314.35037 [7] Chen, C.; Kuo, Y.; Wu, T., The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differ. Equ., 250, 1876-1908 (2011) · Zbl 1214.35077 [8] Cheng, B.; Wu, X., Existence results of positive solutions of Kirchhoff problems, Nonlinear Anal., 71, 4883-4892 (2009) · Zbl 1175.35038 [9] Colasuonno, F.; Pucci, P., Multiplicity of solutions for $$p(x)$$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74, 5962-5974 (2011) · Zbl 1232.35052 [10] Duan, L.; Huang, L. H., Infinitely many solutions for sublinear Schrödinger-Kirchhoff-type equations with general potentials, Results Math., 66, 181-197 (2014) · Zbl 1306.35018 [11] Iannizzotto, A.; Papageorgiou, N., Existence and multiplicity results for resonant fractional boundary value problems, Discrete Contin. Dyn. Syst., Ser. S, 11, 3, 511-532 (2018) · Zbl 1386.35439 [12] Jiu, Q.; Su, J., Existence and multiplicity results for Dirichlet problems with p-Laplacian, J. Math. Anal. Appl., 281, 587-601 (2003) · Zbl 1146.35358 [13] Kirchhoff, G., Mechanik (1883), Leipzig: Teubner, Leipzig [14] Lei, C.-Y.; Liu, G.-S.; Guo, L.-T., Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity, Nonlinear Anal., Real World Appl., 31, 343-355 (2016) · Zbl 1339.35102 [15] Li, A.; Su, J., Existence and multiplicity of solutions for Kirchhoff-type equation with radial potentials in $$\mathbb{R}^3$$, Z. Angew. Math. Phys., 66, 3147-3158 (2015) · Zbl 1332.35111 [16] Li, L.; Zhong, X., Infinitely many small solutions for the Kirchhoff equation with local sublinear nonlinearities, J. Math. Anal. Appl., 435, 955-967 (2016) · Zbl 1328.35026 [17] Liang, Z.; Li, F.; Shi, J., Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 31, 155-167 (2014) · Zbl 1288.35456 [18] Lions, J. L., On some questions in boundary value problems of mathematical physics, Contemporary Developments in Continuum Mechanics and Partial Differential Equations. Proc. Internat. Sympos., Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 284-346 (1978), Amsterdam: North-Holland, Amsterdam [19] Liu, J., Wu, S.: A note on a class of sublinear elliptic equation (1997) preprint, Peking University [20] Mao, A.; Zhang, Z., Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70, 1275-1287 (2009) · Zbl 1160.35421 [21] Moroz, V., Solutions of superlinear at zero elliptic equations via Morse theory, Topol. Methods Nonlinear Anal., 10, 387-397 (1997) · Zbl 0919.35048 [22] Nyamoradi, N.; Zhou, Y., Existence of solutions for a Kirchhoff type fractional differential equations via minimal principle and Morse theory, Topol. Methods Nonlinear Anal., 46, 617-630 (2015) · Zbl 1360.34017 [23] Perera, K.; Zhang, Z., Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ., 221, 1, 246-255 (2006) · Zbl 1357.35131 [24] Sun, J.; Liu, S., Nontrivial solutions of Kirchhoff type problems, Appl. Math. Lett., 25, 500-504 (2012) · Zbl 1251.35027 [25] Sun, J.; Tang, C., Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74, 1212-1222 (2011) · Zbl 1209.35033 [26] Sun, M.; Su, J.; Cai, H., Multiple solutions for the p-Laplacian equations with concave nonlinearities via Morse theory, Commun. Contemp. Math., 19 (2017) · Zbl 1376.35085 [27] Tayyebi, E.; Nyamoradi, N., Existence of nontrivial solutions for Kirchhoff type fractional differential equations with Liouville-Weyl fractional derivatives, Nonlinear Anal., 2018 (2018) [28] Wang, L.; Han, Z., Multiple small solutions for Kirchhoff equation with local sublinear nonlinearities, Appl. Math. Lett., 59, 31-37 (2016) · Zbl 1341.35025 [29] Xiang, M.; Zhang, B.; Guo, X., Infinitely many solutions for a fractional Kirchhoff type problem via fountain theorem, Nonlinear Anal., 120, 299-313 (2015) · Zbl 1328.35287 [30] Yang, Y.; Zhang, J., Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23, 377-380 (2010) · Zbl 1188.35084 [31] Zhang, B.; Bisci, G. M.; Xiang, M., Multiplicity results for nonlocal fractional p-Kirchhoff equations via Morse theory, Topol. Methods Nonlinear Anal., 49, 445-461 (2017) · Zbl 1370.35270 [32] Zhang, Z.; Perera, K., Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317, 456-463 (2006) · Zbl 1100.35008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.