Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces
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Authors
Liben Wang
- Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China.
- Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China.
Xingyong Zhang
- Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China.
Hui Fang
- Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China.
- Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China.
Abstract
In this paper, we investigate the following nonlinear and non-homogeneous elliptic system
\[
\begin{cases}
-div(\phi_1(|\nabla u|)\nabla u)= F_u(x,u,v),\,\,\,\,\, \texttt{in} \Omega,\\
-div(\phi_2(|\nabla v|)\nabla v)= F_v(x,u,v),\,\,\,\,\, \texttt{in} \Omega,\\
u=v=0,\,\,\,\,\, \texttt{on} \partial \Omega.
\end{cases}
\]
where \(\Omega\)
is a bounded domain in \(R^N(N \geq 2)\) with smooth boundary \(\partial\Omega\)
, functions \(\phi_i(t)t (i = 1, 2)\) are increasing homeomorphisms
from \(R^+\) onto \(R^+\). When \(F\) satisfies some \((\phi_1,\phi_2)\)-superlinear and subcritical growth conditions at infinity, by using the
mountain pass theorem we obtain that system has a nontrivial solution, and when \(F\) satisfies an additional symmetric condition,
by using the symmetric mountain pass theorem, we obtain that system has infinitely many solutions. Some of our results extend
and improve those corresponding results in Carvalho et al. [M. L. M. Carvalho, J. V. A. Goncalves, E. D. da Silva, J. Math. Anal.
Appl., 426 (2015), 466–483].
Share and Cite
ISRP Style
Liben Wang, Xingyong Zhang, Hui Fang, Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 7, 3792--3814
AMA Style
Wang Liben, Zhang Xingyong, Fang Hui, Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces. J. Nonlinear Sci. Appl. (2017); 10(7):3792--3814
Chicago/Turabian Style
Wang, Liben, Zhang, Xingyong, Fang, Hui. "Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces." Journal of Nonlinear Sciences and Applications, 10, no. 7 (2017): 3792--3814
Keywords
- Orlicz-Sobolev spaces
- mountain pass theorem
- symmetric mountain theorem.
MSC
References
-
[1]
R. A. Adams, J. F. Fournier, Sobolev spaces, Second edition, Pure and Applied Mathematics (Amsterdam), Elsevier/ Academic Press, Amsterdam (2003)
-
[2]
K. Adriouch, A. El Hamidi, The Nehari manifold for systems of nonlinear elliptic equations, Nonlinear Anal., 64 (2006), 2149–2167.
-
[3]
G. A. Afrouzi, S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the \((p_1, . . . , p_n)\)-Laplacian, Nonlinear Anal., 70 (2009), 135–143.
-
[4]
C. O. Alves, G. M. Figueiredo, J. A. Santos, Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications, Topol. Methods Nonlinear Anal., 44 (2014), 435–456.
-
[5]
G. Anello, On the multiplicity of critical points for parameterized functionals on reflexive Banach spaces, Stud. Math., 213 (2012), 49–60.
-
[6]
P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with ”strong” resonance at infinity, Nonlinear Anal., 7 (1983), 981–1012.
-
[7]
L. Boccardo, D. Guedes de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 309–323.
-
[8]
G. Bonanno, G. Molica Bisci, D. O’Regan, Infinitely many weak solutions for a class of quasilinear elliptic systems, Math. Comput. Modelling, 52 (2010), 152–160.
-
[9]
G. Bonanno, G. Molica Bisci, V. D. Rădulescu , Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz- Sobolev spaces, Nonlinear Anal., 75 (2012), 4441–4456.
-
[10]
Y. Bozhkov, E. Mitidieri , Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations, 190 (2003), 239–267.
-
[11]
F. Cammaroto, L. Vilasi, Multiple solutions for a nonhomogeneous Dirichlet problem in Orlicz-Sobolev spaces, Appl. Math. Comput., 218 (2012), 11518–11527.
-
[12]
M. L. M. Carvalho, J. V. A. Goncalves, E. D. da Silva, On quasilinear elliptic problems without the Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 426 (2015), 466–483.
-
[13]
N. T. Chung, H. Q. Toan, On a nonlinear and non-homogeneous problem without (A-R) type condition in Orlicz-Sobolev spaces, Appl. Math. Comput., 219 (2013), 7820–7829.
-
[14]
P. Clément, M. García-Huidobro, R. Manásevich, K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 11 (2000), 33–62.
-
[15]
P. De Nápoli, M. C. Mariani, Mountain pass solutions to equations of p-Laplacian type, Nonlinear Anal., 54 (2003), 1205–1219.
-
[16]
A. El Khalil, M. Ouanan, A. Touzani, Existence and regularity of positive solutions for an elliptic system, Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf., Southwest Texas State Univ., San Marcos, TX, 9 (2002), 171–182.
-
[17]
F. Fang, Z. Tan , Existence and multiplicity of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting, J. Math. Anal. Appl., 389 (2012), 420–428.
-
[18]
N. Fukagai, M. Ito, K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on \(R^N\), Funkcial. Ekvac., 49 (2006), 235–267.
-
[19]
N. Fukagai, M. Ito, K. Narukawa, Quasilinear elliptic equations with slowly growing principal part and critical Orlicz- Sobolev nonlinear term, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 73–106.
-
[20]
N. Fukagai, K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat. Pura Appl., 186 (2007), 539–564.
-
[21]
M. García-Huidobro, V. K. Le, R. Manásevich, K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 207–225.
-
[22]
J. P. Gossez, Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems, Nonlinear analysis, function spaces and applications, Proc. Spring School, Horni Bradlo, (1978), Teubner, Leipzig, (1979), 59–94.
-
[23]
J. Huentutripay, R. Manásevich, Nonlinear eigenvalues for a quasilinear elliptic system in Orlicz-Sobolev spaces, J. Dynam. Differential Equations, 18 (2006), 901–929.
-
[24]
V. K. Le, Some existence results and properties of solutions in quasilinear variational inequalities with general growths, Differ. Equ. Dyn. Syst., 17 (2009), 343–364.
-
[25]
G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203– 1219.
-
[26]
G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311–361.
-
[27]
J.-J. Liu, X.-Y. Shi, Existence of three solutions for a class of quasilinear elliptic systems involving the (p(x), q(x))-Laplacian, Nonlinear Anal., 71 (2009), 550–557.
-
[28]
M. Mihăilescu, D. Repovš, Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz-Sobolev spaces, Appl. Math. Comput., 217 (2011), 6624–6632.
-
[29]
P. Pucci, J. Serrin, The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1–66.
-
[30]
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1986)
-
[31]
M. M. Rao, Z. D. Ren, Applications of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York (2002)
-
[32]
B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal., 74 (2011), 7446–7454.
-
[33]
J. A. Santos, Multiplicity of solutions for quasilinear equations involving critical Orlicz-Sobolev nonlinear terms, Electron. J. Differential Equations, 2013 (2013), 13 pages.
-
[34]
L.-B. Wang, X.-Y. Zhang, H. Fang, Multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, Taiwanese J. Math., (2017), 32 pages.
-
[35]
T.-F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Anal., 68 (2008), 1733–1745.
-
[36]
F.-L. Xia, G.-X. Wang, Existence of solution for a class of elliptic systems, J. Hunan Agric. Univ. Nat. Sci., 33 (2007), 362–366.
-
[37]
J. F. Zhao, Structure theory of Banach spaces, (Chinese) Wuhan Univ. Press, Wuhan (1991)