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

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.
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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[3] K. C. Chang, Critical Point Theory and Applications, Shanghai Scientific and Technology Press, Shanghai, 1986.   Google Scholar
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Y. ChenS. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[8]

X. FanQ. 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.  Google Scholar

[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.  Google Scholar

[10]

B. GeQ. 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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[27] C. ZhongX. Fan and W. Chen, Introduction to Nonlinear Functional Analysis, Lanzhou University Press, Lanzhou, 1998.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

X. FanQ. 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.  Google Scholar

[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.  Google Scholar

[10]

B. GeQ. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[27] C. ZhongX. Fan and W. Chen, Introduction to Nonlinear Functional Analysis, Lanzhou University Press, Lanzhou, 1998.   Google Scholar
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