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Global attractor for a strongly damped wave equation with fully supercritical nonlinearities
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 |
$\left\{ \begin{align} &-{{\vartriangle }_{p(x)}}u=\lambda \alpha (x)|u{{\text{ }\!\!|\!\!\text{ }}^{\alpha (x)-2}}u|v{{\text{ }\!\!|\!\!\text{ }}^{\beta (x)}}+{{F}_{u}}(x,u,v),\text{ in }\Omega , \\ &-{{\vartriangle }_{q(x)}}v=\lambda \beta (x)\text{ }\!\!|\!\!\text{ }u{{|}^{\alpha \left( x \right)}}|v{{|}^{\beta (x)\text{-2}}}v+{{F}_{v}}(x,u,v),\text{ in}\ \Omega , \\ &u=0=v,\text{ on }\partial \Omega \text{.} \\ \end{align} \right.$ |
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. Google Scholar |
[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. 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. |
[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. 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.
|
[27] | C. Zhong, X. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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.
|
[27] | C. Zhong, X. Fan and W. Chen, Introduction to Nonlinear Functional Analysis, Lanzhou University Press, Lanzhou, 1998. Google Scholar |
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