Hassan Belaouidel     (Department of Mathematics and Computer Science, Faculty of Sciences, University Mohamed I, Mohammed V av., P.O. Box 524, Oujda 60000, Morocco)
Anass Ourraoui     (Department of Mathematics and Computer Science, Faculty of Sciences, University Mohamed I, Mohammed V av., P.O. Box 524, Oujda 60000, Morocco)
Najib Tsouli     (Department of Mathematics and Computer Science, Faculty of Sciences, University Mohamed I, Mohammed V av., P.O. Box 524, Oujda 60000, Morocco)


This paper deals with the existence and multiplicity of solutions for a class of quasilinear problems involving \(p(x)\)-Laplace type equation, namely $$
-\mathrm{div}\, (a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u)= \lambda f(x,u)&\text{in}&\Omega,\\
n\cdot a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u +b(x)|u|^{p(x)-2}u=g(x,u) &\text{on}&\partial\Omega.
Our technical approach is based on variational methods, especially, the mountain pass theorem and the symmetric mountain pass theorem.


\(p(x)\)-Laplacian, Mountain pass theorem, Multiple solutions, Critical point theory.

Full Text:



Allaoui M., El Amrouss A., Ourraoui A. Existence of infinitely many solutions for a Steklov problem involving the \(p(x)\)-Laplace operator. Electron. J. Qual. Theory Differ. Equ., 2014. No. 20. P. 1–10. DOI: 10.14232/ejqtde.2014.1.20

Antontsev S, Shmarev S., Chapter 1. Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions. In: Handbook of Differential Equations, Stationary Partial Differ. Equ. Chipot M., Quittner P. (eds.), 2006. Vol. 3. P. 1–100. DOI: 10.1016/S1874-5733(06)80005-7

Bocea M., Mihǎilescu M. \(\Gamma\)-convergence of power-law functionals with variable exponents. Nonlinear Anal.: Theory, Methods, Appl., 2010. Vol. 73, No. 1. P. 110–121. DOI: 10.1016/

Bocea M., Mihǎilescu M., Popovici C. On the asymptotic behavior of variable exponent power-law functionals and applications. Ric. Mat., 2010. Vol. 59. P. 207–238. DOI: 10.1007/s11587-010-0081-x

Chabrowski J., Fu Y. Existence of solutions for \(p(x)\)-Laplacian problems on a bounded domain. J. Math. Anal. Appl., 2005. Vol. 306, No. 2. P. 604–618. DOI: 10.1016/j.jmaa.2004.10.028

Chen Y., Levine S., Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math., 2006. Vol. 66, No. 2. P. 1383–1406. DOI: 10.1137/050624522

Dai G. Infinitely many solutions for a \(p(x)\)-Laplacian equation in \(\mathbb{R}^N\). Nonlinear Anal.: Theory, Methods, Appl., 2009. Vol. 71, No. 3–4. P. 1133–1139. DOI: 10.1016/

Deng S.-G. Positive solutions for Robin problem involving the \(p(x)\)-Laplacian. J. Math. Anal. Appl., 209 Vol. 360, No. 2. P. 548–560. DOI: 10.1016/j.jmaa.2009.06.032

Deng S.-G. Eigenvalues of the \(p(x)\)-Laplacian Steklov problem. J. Math. Anal. Appl., 2008. Vol. 339, No. 2. P. 925–937. DOI: 10.1016/j.jmaa.2007.07.028

Deng S.-G. A local mountain pass theorem and applications to a double perturbed \(p(x)\)-Laplacian equations. Appl. Math. Comput., 2009. Vol. 211, No. 1. P. 234–241. DOI: 10.1016/j.amc.2009.01.042

Diening L., Hästö P., Nekvinda A. Open problems in variable exponent Lebesgue and Sobolev spaces. In: Function Spaces, Differential Operators And Nonlinear Analysis. Proc. Conference Held in Milovy, Bohemian-Moravian Uplands. Drábek P., Rákosník J. (Eds.) May 28 – June 2, 2004, Milovy, Czech Republic. Milovy: Math. Inst. Acad. Sci. Czech, 2005. P. 38–58. URL:

Diening L., Harjulehto P., Hästö P., Ruzicka M. Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Math., vol. 2017. Berlin, Heidelberg: Springer-Verlag, 2011. 509 p. DOI: 10.1007/978-3-642-18363-8

Edmunds D.E., Rákosník J. Sobolev embeddings with variable exponent. Studia Math., 2000. Vol. 143, No. 3. P. 267–293.

El Amrouss A., Moradi F., Ourraoui A. Neumann problem in divergence form modeled on the p(x)-Laplace equation. Bol. Soc. Parana. Mat. (3), 2014. Vol. 32, No. 2. P. 109–117. DOI: 10.5269/bspm.v32i2.20329

Fan X., Han X. Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(\mathbb{R}^N\). Nonlinear Anal.: Theory Methods Appl., 2004. Vol. 59. P. 173–188. DOI: 10.1016/

Fan X., Zhao D. On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m,p(x)}(\Omega)\). J. Math. Anal. Appl., 2001. Vol. 263, No. 2. P. 424–446. DOI: 10.1006/jmaa.2000.7617

Fan X.-L., Zhang Q.-H. Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. Nonlinear Anal.: Theory Methods Appl., 2003. Vol. 52, No. 8. P. 1843–1852. DOI: 10.1016/S0362-546X(02)00150-5

Fu Y., Zhang X. A multiplicity result for \(p(x)\)-Laplacian problem in \(\mathbb{R}^N\). Nonlinear Anal.: Theory, Methods, Appl., 2009. Vol. 70, No. 6. P. 2261–2269. DOI: 10.1016/

Juárez Hurtado E., Miyagaki O.H., Rodrigues R.S. Existence and Multiplicity of Solutions for a Class of Elliptic Equations Without Ambrosetti–Rabinowitz Type Conditions. J. Dynam. Differential Equations, 2018. Vol. 30. P. 405–432. DOI: 10.1007/s10884-016-9542-6

Kováčik O., Rákosník J. On spaces \(L^{ p(x)}\) and \(W^{k,p(x)}\). Czechoslovak Math. J., 1991. Vol. 41, No. 4. P. 592–618. URL:

Mihǎilescu M. On a class of nonlinear problems involving a \(p(x)\)-Laplace type operator. Czechoslovak Math. J., 2008. Vol. 58, No. 1. P. 155–172. URL:

Ni W.-M., Serrin J. Existence and non-existence theorems for ground states for quasilinear partial differential equations. Att. Conveg. Lincei, 1985. Vol. 7. P. 231–257.

Ourraoui A. Multiplicity results for Steklov problem with variable exponent. Appl. Math. Comput., 2016. Vol. 277. P. 34–43. DOI: 10.1016/j.amc.2015.12.043

Ourraoui A. Some Results for Robin Type Problem Involving \(p(x)\)-Laplacian. Preprint.

Pflüger K. Existence and multiplicity of solutions to a \(p\)-Laplacian equation with nonlinear boundary condition. Electron. J. Differential Equations, 1998. Vol. 1998, No. 10. P. 1–13.

Rabinowitz P.H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics, Vol. 65. Providence, Rhode Island: American Mathematical Soceity, 1986. 100 p.

Růžička M. Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Math., vol. 1748. Berlin, Heidelberg: Springer-Verlag, 2002. 178 p. DOI: 10.1007/BFb0104029

Silva E.A.B., Xavier M.S. Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2003. Vol. 20, No. 2. P. 341–358. DOI: 10.1016/S0294-1449(02)00013-6

Zhao D., Fan X.L. On the Nemytskiǐ operators from \(L^{p_1(x)}(\Omega)\) to \(L^{p_2(x)}(\Omega)\). J. Lanzhou Univ. Nat. Sci., 1998. Vol. 34, No. 1. P. 1–5. (in Chinese)

Zhao J.F. Structure Theory of Banach Spaces. Wuhan: Wuhan Univ. Press, 1991. (in Chinese)

Zhou Q.-M., Ge B. Multiple solutions for a Robin-type differential inclusion problem involving the \(p(x)\)- Laplacian. Math. Methods Appl. Sci., 2013. Vol. 40, No. 18. P. 6229–6238. DOI: 10.1002/mma.2760

Zhikov V.V. Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR-Izv. , 1987. Vol. 29, No. 1. P. 33–66. DOI: 10.1070/IM1987v029n01ABEH000958


Article Metrics

Metrics Loading ...


  • There are currently no refbacks.