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April  2017, 37(4): 2207-2226. doi: 10.3934/dcds.2017095

Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth

Li Yin 1, , Jinghua Yao 2,, , Qihu Zhang 3, and  Chunshan Zhao 4,

1. 

College of Information and Management Science, Henan Agricultural University, Zhengzhou, Henan 450002, China
2. 

Department of Mathematics, Indiana University, Bloomington, IN 47408, USA
3. 

Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China
4. 

Department of Mathematical Science, Georgia Southern University, Statesboro, GA 30460, USA

*Corresponding author: Jinghua Yao

Received  July 2016 Revised  September 2016 Published  December 2016

Fund Project: This research is partly supported by the key projects in Science and Technology Research of the Henan Education Department (14A110011).

We investigate the followingDirichlet problem with variable exponents:
$\left\{ \begin{align} &-{{\vartriangle }_{p(x)}}u=\lambda \alpha (x)|u{{\text{ }\!\!|\!\!\text{ }}^{\alpha (x)-2}}u|v{{\text{ }\!\!|\!\!\text{ }}^{\beta (x)}}+{{F}_{u}}(x,u,v),\text{ in }\Omega , \\ &-{{\vartriangle }_{q(x)}}v=\lambda \beta (x)\text{ }\!\!|\!\!\text{ }u{{|}^{\alpha \left( x \right)}}|v{{|}^{\beta (x)\text{-2}}}v+{{F}_{v}}(x,u,v),\text{ in}\ \Omega , \\ &u=0=v,\text{ on }\partial \Omega \text{.} \\ \end{align} \right.$
We present here, in the system setting, a new set of growth conditions under which we manage to use a novel method to verify the Cerami compactness condition. By localization argument, decomposition technique and variational methods, we are able to show the existence of multiple solutions with constant sign for the problem without the well-knownAmbrosetti-Rabinowitz type growth condition. More precisely, we manage to show that the problem admitsfour, six and infinitely many solutions respectively.
Keywords: $ p(x) $-Laplacian, Dirichlet problem, solutions with constant sign, Ambrosetti-Rabinowitz condition, Cerami condition, critical point.
Mathematics Subject Classification: Primary:35J20, 35J25;Secondary:35J60.
Citation: Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao. Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2207-2226. doi: 10.3934/dcds.2017095
References:
[1]

E. Acerbi and G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140.  doi: 10.1007/s002050100117.

[2]

C. Alves and S. Liu, On superlinear $ p(x) $-Laplacian equations in $ R^{N} $, Nonlinear Analysis, 73 (2010), 2566-2579.  doi: 10.1016/j.na.2010.06.033.

[3] K. C. Chang, Critical Point Theory and Applications, Shanghai Scientific and Technology Press, Shanghai, 1986. 
[4]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.

[5]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-18363-8.

[6]

X. Fan and D. Zhao, On the spaces $ {{L}^{p(x)}}(\Omega \text{)} $ and $ {{W}^{m,p(x)}}\left( \Omega \right) $, J. Math. Anal. Appl., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.

[7]

X. Fan and Q. Zhang, Existence of solutions for $ p(x) $-Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5.

[8]

X. Fan, Q. Zhang and D. Zhao, Eigenvalues of $ p(x) $-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306-317.  doi: 10.1016/j.jmaa.2003.11.020.

[9]

L. Gasiński and N. Papageorgiou, A pair of positive solutions for the Dirichlet $ p(z) $-Laplacian with concave and convex nonlinearities, J. Glob. Optim., 56 (2013), 1347-1360.  doi: 10.1007/s10898-011-9841-8.

[10]

B. Ge, Q. Zhou and L. Zu, Positive solutions for nonlinear elliptic problems of $ p $-Laplacian type on $ \mathbb{R}^{N} $ without (AR) condition, Nonlinear Anal Real World Appl., 21 (2015), 99-109.  doi: 10.1016/j.nonrwa.2014.07.002.

[11]

O. Kováčik and J. Rákosník, On spaces $ {{L}^{p(x)}}\left( \Omega \right) $ and $ {{W}^{k,p(x)}}\left( \Omega \right) $, Czechoslovak Math. J., 41 (1991), 592-618. 

[12]

N. Lam and G. Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, J. Geom. Anal., 24 (2014), 118-143.  doi: 10.1007/s12220-012-9330-4.

[13]

M. Mihăilescu and V. Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., 135 (2007), 2929-2937.  doi: 10.1090/S0002-9939-07-08815-6.

[14]

O. Miyagaki and M. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations, 245 (2008), 3628-3638.  doi: 10.1016/j.jde.2008.02.035.

[15]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[16]

V. Radulescu and D. Repovs, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Chapman and Hall/CRC, Monographs and Research Notes in Mathematics, 2015. doi: 10.1201/b18601.

[17]

X. Wang, J. Yao and D. Liu, High energy solutions to $ p(x) $-Laplace equations of Schrödinger type, Electron. J. Diff. Equ., 136 (2015), 1-17. 

[18]

X. Wang and J. Yao, Compact embeddings between variable exponent spaces with unbounded underlying domain, Nonlinear Analysis: TMA, 70 (2009), 3472-3482.  doi: 10.1016/j.na.2008.07.005.

[19]

M. Willem and W. Zou, On a Schröinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J., 52 (2003), 109-132.  doi: 10.1512/iumj.2003.52.2273.

[20]

J. Yao and X. Wang, On an open problem involving the $ p(x) $-Laplacian, Nonlinear Analysis: TMA, 69 (2008), 1445-1453.  doi: 10.1016/j.na.2007.06.044.

[21]

J. Yao, Solutions for Neumann boundary value problems involving $ p(x) $-Laplace operators, Nonlinear Analysis: TMA, 68 (2008), 1271-1283.  doi: 10.1016/j.na.2006.12.020.

[22]

L. Yin, J. Yao, Q. Zhang and C. Zhao, Multiplicity of strong solutions for a class of elliptic problems without the Ambrosetti-Rabinowitz condition in $ \mathbb{R}^{N} $, arXiv: 1607.00581.

[23]

A. Zang, $ p(x) $-Laplacian equations satisfying Cerami condition, J. Math. Anal. Appl., 337 (2008), 547-555.  doi: 10.1016/j.jmaa.2007.04.007.

[24]

Q. Zhang and C. Zhao, Existence of strong solutions of a $ p(x) $-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comput. Math. Appl., 69 (2015), 1-12.  doi: 10.1016/j.camwa.2014.10.022.

[25] J. Zhao, Structure Theory of Banach Spaces, Wuhan University Press, Wuhan, 1991. 
[26]

V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710. 

[27] C. Zhong, X. Fan and W. Chen, Introduction to Nonlinear Functional Analysis, Lanzhou University Press, Lanzhou, 1998. 

show all references

References:
[1]

E. Acerbi and G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140.  doi: 10.1007/s002050100117.

[2]

C. Alves and S. Liu, On superlinear $ p(x) $-Laplacian equations in $ R^{N} $, Nonlinear Analysis, 73 (2010), 2566-2579.  doi: 10.1016/j.na.2010.06.033.

[3] K. C. Chang, Critical Point Theory and Applications, Shanghai Scientific and Technology Press, Shanghai, 1986. 
[4]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.

[5]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-18363-8.

[6]

X. Fan and D. Zhao, On the spaces $ {{L}^{p(x)}}(\Omega \text{)} $ and $ {{W}^{m,p(x)}}\left( \Omega \right) $, J. Math. Anal. Appl., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.

[7]

X. Fan and Q. Zhang, Existence of solutions for $ p(x) $-Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5.

[8]

X. Fan, Q. Zhang and D. Zhao, Eigenvalues of $ p(x) $-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306-317.  doi: 10.1016/j.jmaa.2003.11.020.

[9]

L. Gasiński and N. Papageorgiou, A pair of positive solutions for the Dirichlet $ p(z) $-Laplacian with concave and convex nonlinearities, J. Glob. Optim., 56 (2013), 1347-1360.  doi: 10.1007/s10898-011-9841-8.

[10]

B. Ge, Q. Zhou and L. Zu, Positive solutions for nonlinear elliptic problems of $ p $-Laplacian type on $ \mathbb{R}^{N} $ without (AR) condition, Nonlinear Anal Real World Appl., 21 (2015), 99-109.  doi: 10.1016/j.nonrwa.2014.07.002.

[11]

O. Kováčik and J. Rákosník, On spaces $ {{L}^{p(x)}}\left( \Omega \right) $ and $ {{W}^{k,p(x)}}\left( \Omega \right) $, Czechoslovak Math. J., 41 (1991), 592-618. 

[12]

N. Lam and G. Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, J. Geom. Anal., 24 (2014), 118-143.  doi: 10.1007/s12220-012-9330-4.

[13]

M. Mihăilescu and V. Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., 135 (2007), 2929-2937.  doi: 10.1090/S0002-9939-07-08815-6.

[14]

O. Miyagaki and M. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations, 245 (2008), 3628-3638.  doi: 10.1016/j.jde.2008.02.035.

[15]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[16]

V. Radulescu and D. Repovs, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Chapman and Hall/CRC, Monographs and Research Notes in Mathematics, 2015. doi: 10.1201/b18601.

[17]

X. Wang, J. Yao and D. Liu, High energy solutions to $ p(x) $-Laplace equations of Schrödinger type, Electron. J. Diff. Equ., 136 (2015), 1-17. 

[18]

X. Wang and J. Yao, Compact embeddings between variable exponent spaces with unbounded underlying domain, Nonlinear Analysis: TMA, 70 (2009), 3472-3482.  doi: 10.1016/j.na.2008.07.005.

[19]

M. Willem and W. Zou, On a Schröinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J., 52 (2003), 109-132.  doi: 10.1512/iumj.2003.52.2273.

[20]

J. Yao and X. Wang, On an open problem involving the $ p(x) $-Laplacian, Nonlinear Analysis: TMA, 69 (2008), 1445-1453.  doi: 10.1016/j.na.2007.06.044.

[21]

J. Yao, Solutions for Neumann boundary value problems involving $ p(x) $-Laplace operators, Nonlinear Analysis: TMA, 68 (2008), 1271-1283.  doi: 10.1016/j.na.2006.12.020.

[22]

L. Yin, J. Yao, Q. Zhang and C. Zhao, Multiplicity of strong solutions for a class of elliptic problems without the Ambrosetti-Rabinowitz condition in $ \mathbb{R}^{N} $, arXiv: 1607.00581.

[23]

A. Zang, $ p(x) $-Laplacian equations satisfying Cerami condition, J. Math. Anal. Appl., 337 (2008), 547-555.  doi: 10.1016/j.jmaa.2007.04.007.

[24]

Q. Zhang and C. Zhao, Existence of strong solutions of a $ p(x) $-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comput. Math. Appl., 69 (2015), 1-12.  doi: 10.1016/j.camwa.2014.10.022.

[25] J. Zhao, Structure Theory of Banach Spaces, Wuhan University Press, Wuhan, 1991. 
[26]

V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710. 

[27] C. Zhong, X. Fan and W. Chen, Introduction to Nonlinear Functional Analysis, Lanzhou University Press, Lanzhou, 1998. 
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