# American Institute of Mathematical Sciences

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

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

 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.
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 - A, 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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 (1978), 284. 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. 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. 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, (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. Google Scholar [17] J. J. Sun and C. L. Tang, Resonance problems for Kirchhoff type equations,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 2139. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.jmaa.2005.06.102. Google Scholar

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##### 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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 (1978), 284. 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. 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. 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, (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. Google Scholar [17] J. J. Sun and C. L. Tang, Resonance problems for Kirchhoff type equations,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 2139. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.jmaa.2005.06.102. Google Scholar
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