Existence and Multiplicity of Solutions for a Robin Problem
-
2891
Downloads
-
4167
Views
Authors
Mostafa Allaoui
- Department of Mathematics, University Mohamed I, Oujda, Morocco.
Abdel Rachid El Amrouss
- Department of Mathematics, University Mohamed I, Oujda, Morocco.
Fouad Kissi
- Department of Mathematics, University Mohamed I, Oujda, Morocco.
Anass Ourraoui
- Department of Mathematics, University Mohamed I, Oujda, Morocco.
Abstract
In this article we study the nonlinear Robin boundary-value problem
\[
\begin{cases}
-\Delta_{p(x)}u=\lambda f(x,u),\,\,\,\,\, \texttt{in}\quad \Omega,\\
|\nabla u|^{p(x)-2} \frac {\partial u}{\partial v} + \beta(x)|u|^{p(x)-2} u=0,\,\,\,\,\, \texttt{on}\quad \partial \Omega.
\end{cases}
\]
Using the variational method, under appropriate assumptions on \(f\), we obtain a result on existence and multiplicity of solutions.
Share and Cite
ISRP Style
Mostafa Allaoui, Abdel Rachid El Amrouss, Fouad Kissi, Anass Ourraoui, Existence and Multiplicity of Solutions for a Robin Problem, Journal of Mathematics and Computer Science, 10 (2014), no. 3, 163-172
AMA Style
Allaoui Mostafa, Amrouss Abdel Rachid El, Kissi Fouad, Ourraoui Anass, Existence and Multiplicity of Solutions for a Robin Problem. J Math Comput SCI-JM. (2014); 10(3):163-172
Chicago/Turabian Style
Allaoui, Mostafa, Amrouss, Abdel Rachid El, Kissi, Fouad, Ourraoui, Anass. "Existence and Multiplicity of Solutions for a Robin Problem." Journal of Mathematics and Computer Science, 10, no. 3 (2014): 163-172
Keywords
- \(p(x)\)-Laplace operator
- variable exponent Lebesgue space
- variable exponent Sobolev space
- Riccerifs variational principle.
MSC
References
-
[1]
G. A. Afrouzi, T. N . Ghara, Existence of three weak solutions for elliptic Dirichlet problem, The Journal of mathematics and computer Science, 3 (2012), 386-391.
-
[2]
G. A. Afrouzi, S. Shamilo, M. Mahdavi, Three solution for a class of quasilinear Dirichlet elliptic systems involving \((p,q)\)-Laplcian operator, The Journal of mathematics and computer Science, 3 (2012), 487-493.
-
[3]
M. Allaoui, A. R El Amrouss, Solutions for Steklov boundary value problems involving \(p(x)\)-Laplace operators, Bol. Soc. Paran. Mat. , 32 (2014), 163-173.
-
[4]
M. Allaoui, A. R. El. Amrouss, A. Ourraoui, Three solutions for a quasi-linear elliptic problem, Applied Mathematics E-Notes, 13 (2013), 51-59.
-
[5]
G. Bonanno, P. Candito, Three solutions to a Neumann problem for elliptic equations involving the \(p\)-Laplacian , Arch. Math. (Basel), 80 (2003), 424-429.
-
[6]
J. Chabrowski, Y. Fu, Existence of solutions for \(p(x)\)-Laplacian problems on a bounded domain , J. Math. Anal. Appl., 306 (2005), 604-618.
-
[7]
Y. M. Chen, S. Levine, M. Ra, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. , 66 (2006), 1383-1406.
-
[8]
S. G. Dend, Qin Wang, Shijuan Cheng, On the p(x)-Laplacian Robin eigenvalue problem, Appl. Math. Comput., 217 (2011), 5643-5649.
-
[9]
S. G. Deng, A local mountain pass theorem and applications to a double perturbed \(p(x)\)-Laplacian equations, Appl. Math. Comput., 211 (2009), 234-241.
-
[10]
S. G. Deng, Positive solutions for Robin problem involving the \(p(x)\)-Laplacian, J. Math. Anal. Appl. , 360 (2009), 548-560.
-
[11]
X. Ding, X. Shi , Existence and multiplicity of solutions for a general \(p(x)\)-laplacian Neumann problem, Nonlinear. Anal., 70 (2009), 3715-3720.
-
[12]
X. L. Fan, S.-G. Deng, Remarks on Ricceri’s variational principle and applications to the \(p(x)\)-Laplacian equations, Nonlinear Anal. , 67 (2007), 3064-3075.
-
[13]
X. L. Fan, J. S. Shen, D. Zhao, Sobolev embedding theorems for spaces \(W^{k,p(x)}\), J. Math. Anal. Appl., 262 (2001), 749-760.
-
[14]
X. L. Fan, D. Zhao, On the spaces \(L^{p(x)}\) and \(W^{m,p(x)}\), J. Math. Anal. Appl. , 263 (2001), 424-446.
-
[15]
C. Ji, Remarks on the existence of three solutions for the \(p(x)\)-Laplacian equations, Nonlinear Anal., 74 (2011), 2908-2915.
-
[16]
T. G. Myers, Thin films with high surface tension, SIAM Review, 40 (3) (1998), 441-462.
-
[17]
B. Ricceri, On three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226.
-
[18]
M. Ružicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin (2000)
-
[19]
L. L. Wang, Y. H. Fan, W. G. Ge, Existence and multiplicity of solutions for a Neumann problem involving the \(p(x)\)-laplace operator, Nonlinear. Anal., 71 (2009), 4259-4270.
-
[20]
J. H. Yao, Solution for Neumann boundary problems involving \(p(x)\)-Laplace operators, Nonlinear Anal. , 68 (2008), 1271-1283.
-
[21]
Q. H. Zhang, Existence of solutions for \(p(x)\)-Laplacian equations with singular coefficients in \(R^N\), J. Math. Anal. Appl. , 348 (2008), 38-50.
-
[22]
V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. , 29 (1987), 33-66.
-
[23]
V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, translated from Russian by G. A. Yosifian, Springer-Verlag, Berlin (1994)