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

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.  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

[1]

Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040

[2]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[3]

Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020287

[4]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[5]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229

[6]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395

[7]

Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101

[8]

Xianwei Chen, Xiangling Fu, Zhujun Jing. Chaos control in a special pendulum system for ultra-subharmonic resonance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 847-860. doi: 10.3934/dcdsb.2020144

[9]

Helin Guo, Huan-Song Zhou. Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1023-1050. doi: 10.3934/dcds.2020308

[10]

Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021018

[11]

Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020297

[12]

Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020292

[13]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[14]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034

[15]

Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045

[16]

Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230

[17]

Tomáš Bodnár, Philippe Fraunié, Petr Knobloch, Hynek Řezníček. Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 785-801. doi: 10.3934/dcdss.2020333

[18]

Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263

[19]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[20]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]