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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

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 & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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L. Ding, L. Li and J. L. Zhang, Solutions to Kirchhoff equations with combined nonlinearities, Electron. J. Differential Equations, 2014, No. 10, 10 pp.  Google Scholar

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G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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S. I. Pohožaev, A certain class of quasilinear hyperbolic equations (Russian), Mat. Sb. (N.S.), 96 (1975), 152-166, 168.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. W. Ye, Infinitely many solutions for Kirchhoff type problems, Differ. Equ. Appl., 5 (2013), 83-92. doi: 10.7153/dea-05-06.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations (Russian), Mat. Sb. (N.S.), 96 (1975), 152-166, 168.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. W. Ye, Infinitely many solutions for Kirchhoff type problems, Differ. Equ. Appl., 5 (2013), 83-92. doi: 10.7153/dea-05-06.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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