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
[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

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
[1]

Alessio Fiscella. Schrödinger–Kirchhoff–Hardy $ p $–fractional equations without the Ambrosetti–Rabinowitz condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1993-2007. doi: 10.3934/dcdss.2020154

[2]

Vincenzo Ambrosio. Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2265-2284. doi: 10.3934/dcds.2017099

[3]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003

[4]

Eun Bee Choi, Yun-Ho Kim. Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition. Conference Publications, 2015, 2015 (special) : 276-286. doi: 10.3934/proc.2015.0276

[5]

Lujuan Yu. The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2621-2637. doi: 10.3934/dcdsb.2020025

[6]

Aleksander Denisiuk. On range condition of the tensor x-ray transform in $ \mathbb R^n $. Inverse Problems & Imaging, 2020, 14 (3) : 423-435. doi: 10.3934/ipi.2020020

[7]

Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020021

[8]

K. D. Chu, D. D. Hai. Positive solutions for the one-dimensional singular superlinear $ p $-Laplacian problem. Communications on Pure & Applied Analysis, 2020, 19 (1) : 241-252. doi: 10.3934/cpaa.2020013

[9]

Xin-Guang Yang, Marcelo J. D. Nascimento, Maurício L. Pelicer. Uniform attractors for non-autonomous plate equations with $ p $-Laplacian perturbation and critical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1937-1961. doi: 10.3934/dcds.2020100

[10]

Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1241-1257. doi: 10.3934/dcds.2010.27.1241

[11]

Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070

[12]

Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026

[13]

Nikolay Dimitrov, Stepan Tersian. Existence of homoclinic solutions for a nonlinear fourth order $ p $-Laplacian difference equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 555-567. doi: 10.3934/dcdsb.2019254

[14]

Junyong Eom, Ryuichi Sato. Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4373-4386. doi: 10.3934/cpaa.2020199

[15]

Mihai Mihăilescu, Julio D. Rossi. Monotonicity with respect to $ p $ of the First Nontrivial Eigenvalue of the $ p $-Laplacian with Homogeneous Neumann Boundary Conditions. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4363-4371. doi: 10.3934/cpaa.2020198

[16]

Jiaoxiu Ling, Zhan Zhou. Positive solutions of the discrete Robin problem with $ \phi $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020338

[17]

Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129

[18]

Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062

[19]

Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252

[20]

Changchun Liu, Pingping Li. Global existence for a chemotaxis-haptotaxis model with $ p $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1399-1419. doi: 10.3934/cpaa.2020070

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (43)
  • HTML views (52)
  • Cited by (4)

Other articles
by authors

[Back to Top]