Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues

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November 2016, 36(11): 6453-6473. doi: 10.3934/dcds.2016078

Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues

Shu-Zhi Song ^1, , Shang-Jie Chen ^2, and Chun-Lei Tang ^1,

1.	School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2.	School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received November 2014 Revised May 2016 Published August 2016

Abstract
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We study the following Kirchhoff type problem: \begin{equation*} \left\{ \begin{array}{ccc} -\left(a+b\int_{\Omega}|\nabla u|^2dx \right) \Delta u=f(x,u), &\mbox{in} \ \ \Omega, \\ u=0, &\text{on} \ \partial \Omega. \end{array} \right. \end{equation*} Note that $F(x,t)=\int_0^1 f(x,s)ds$ is the primitive function of $f$. In the first result, we prove the existence of solutions by applying the $G-$Linking Theorem when the quotient $\frac{4F(x,t)}{bt^4}$ stays between $\mu_k$ and $\mu_{k+1}$ allowing for resonance with $\mu_{k+1}$ at infinity. In the second result, for the case that the quotient $\frac{4F(x,t)}{bt^4}$ stays between $\mu_1$ and $\mu'_{2}$ allowing for resonance with $\mu'_{2}$ at infinity, we find a nontrivial solution by using the classical Linking Theorem and argument of the characterization of $\mu'_2$. Meanwhile, similar results are obtained for degenerate problem.

Keywords: Kirchhoff type, link., resonance, Cerami condition, nonlinear eigenvalues.

Mathematics Subject Classification: 35J60, 35A1.

Citation: Shu-Zhi Song, Shang-Jie Chen, Chun-Lei Tang. Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6453-6473. doi: 10.3934/dcds.2016078

References:

[1]	P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong'' resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012. doi: 10.1016/0362-546X(83)90115-3.
[2]	S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées partielles (Russian) Bull. Acad. Sci. URSS, Sér. Math. [Izvestia Akad. Nauk SSSR], 4 (1940), 17-26.
[3]	B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495. doi: 10.1016/j.jmaa.2012.04.025.
[4]	B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892. doi: 10.1016/j.na.2009.03.065.
[5]	M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627. doi: 10.1016/S0362-546X(97)00169-7.
[6]	M. Cuesta, D. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian, J. Differential Equations, 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645.
[7]	P. Drábek and S. B. Robinson, Resonance problems for the $p$-Laplacian, J. Funct. Anal., 169 (1999), 189-200. doi: 10.1006/jfan.1999.3501.
[8]	L. Ding, L. Li and J. L. Zhang, Solutions to Kirchhoff equations with combined nonlinearities, Electron. J. Differential Equations, 2014, No. 10, 10 pp.
[9]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[10]	P. Kanishka and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255. doi: 10.1016/j.jde.2005.03.006.
[11]	Z. P. Liang, F. Y. Li and J. P. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 155-167. doi: 10.1016/j.anihpc.2013.01.006.
[12]	J.-L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary developments in continuum mechanics and partial differential equations, 30, 284-346, North-Holland Math. Stud., North-Holland, Amsterdam-New York, 1978
[13]	A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011.
[14]	A. M. Mao and S. X. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243. doi: 10.1016/j.jmaa.2011.05.021.
[15]	S. Michael, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second edition, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03212-1.
[16]	S. I. Pohožaev, A certain class of quasilinear hyperbolic equations (Russian), Mat. Sb. (N.S.), 96 (1975), 152-166, 168.
[17]	J. J. Sun and C. L. Tang, Resonance problems for Kirchhoff type equations, Discrete Contin. Dyn. Syst., 33 (2013), 2139-2154. doi: 10.3934/dcds.2013.33.2139.
[18]	J. Sun and S. B. Liu, Nontrivial solutions of Kirchhoff type problems, Appl. Math. Lett., 25 (2012), 500-504. doi: 10.1016/j.aml.2011.09.045.
[19]	J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061.
[20]	S. Z. Song and C. L. Tang, Resonance problems for the $p$-Laplacian with a nonlinear boundary condition, Nonlinear Anal., 64 (2006), 2007-2021. doi: 10.1016/j.na.2005.07.035.
[21]	Y. W. Ye, Infinitely many solutions for Kirchhoff type problems, Differ. Equ. Appl., 5 (2013), 83-92. doi: 10.7153/dea-05-06.
[22]	Y. Yang and J. H. Zhang, Positive and negative solutions of a class of nonlocal problems, Nonlinear Anal., 73 (2010), 25-30. doi: 10.1016/j.na.2010.02.008.
[23]	Y. Yang and J. H. Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23 (2010), 377-380. doi: 10.1016/j.aml.2009.11.001.
[24]	Z. T. Zhang and P. Kanishka, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463. doi: 10.1016/j.jmaa.2005.06.102.

show all references

References:

[1]	P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong'' resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012. doi: 10.1016/0362-546X(83)90115-3.
[2]	S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées partielles (Russian) Bull. Acad. Sci. URSS, Sér. Math. [Izvestia Akad. Nauk SSSR], 4 (1940), 17-26.
[3]	B. T. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495. doi: 10.1016/j.jmaa.2012.04.025.
[4]	B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71 (2009), 4883-4892. doi: 10.1016/j.na.2009.03.065.
[5]	M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627. doi: 10.1016/S0362-546X(97)00169-7.
[6]	M. Cuesta, D. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian, J. Differential Equations, 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645.
[7]	P. Drábek and S. B. Robinson, Resonance problems for the $p$-Laplacian, J. Funct. Anal., 169 (1999), 189-200. doi: 10.1006/jfan.1999.3501.
[8]	L. Ding, L. Li and J. L. Zhang, Solutions to Kirchhoff equations with combined nonlinearities, Electron. J. Differential Equations, 2014, No. 10, 10 pp.
[9]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[10]	P. Kanishka and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255. doi: 10.1016/j.jde.2005.03.006.
[11]	Z. P. Liang, F. Y. Li and J. P. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 155-167. doi: 10.1016/j.anihpc.2013.01.006.
[12]	J.-L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary developments in continuum mechanics and partial differential equations, 30, 284-346, North-Holland Math. Stud., North-Holland, Amsterdam-New York, 1978
[13]	A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011.
[14]	A. M. Mao and S. X. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243. doi: 10.1016/j.jmaa.2011.05.021.
[15]	S. Michael, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second edition, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03212-1.
[16]	S. I. Pohožaev, A certain class of quasilinear hyperbolic equations (Russian), Mat. Sb. (N.S.), 96 (1975), 152-166, 168.
[17]	J. J. Sun and C. L. Tang, Resonance problems for Kirchhoff type equations, Discrete Contin. Dyn. Syst., 33 (2013), 2139-2154. doi: 10.3934/dcds.2013.33.2139.
[18]	J. Sun and S. B. Liu, Nontrivial solutions of Kirchhoff type problems, Appl. Math. Lett., 25 (2012), 500-504. doi: 10.1016/j.aml.2011.09.045.
[19]	J. J. Sun and C. L. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222. doi: 10.1016/j.na.2010.09.061.
[20]	S. Z. Song and C. L. Tang, Resonance problems for the $p$-Laplacian with a nonlinear boundary condition, Nonlinear Anal., 64 (2006), 2007-2021. doi: 10.1016/j.na.2005.07.035.
[21]	Y. W. Ye, Infinitely many solutions for Kirchhoff type problems, Differ. Equ. Appl., 5 (2013), 83-92. doi: 10.7153/dea-05-06.
[22]	Y. Yang and J. H. Zhang, Positive and negative solutions of a class of nonlocal problems, Nonlinear Anal., 73 (2010), 25-30. doi: 10.1016/j.na.2010.02.008.
[23]	Y. Yang and J. H. Zhang, Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23 (2010), 377-380. doi: 10.1016/j.aml.2009.11.001.
[24]	Z. T. Zhang and P. Kanishka, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463. doi: 10.1016/j.jmaa.2005.06.102.

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