GENERAL QUASILINEAR PROBLEMS INVOLVING \(p(x)\)-LAPLACIAN WITH ROBIN BOUNDARY CONDITION
Abstract
This paper deals with the existence and multiplicity of solutions for a class of quasilinear problems involving \(p(x)\)-Laplace type equation, namely $$
\left\{\begin{array}{lll}
-\mathrm{div}\, (a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u)= \lambda f(x,u)&\text{in}&\Omega,\\
n\cdot a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u +b(x)|u|^{p(x)-2}u=g(x,u) &\text{on}&\partial\Omega.
\end{array}\right.
$$
Our technical approach is based on variational methods, especially, the mountain pass theorem and the symmetric mountain pass theorem.
Keywords
Full Text:
PDFReferences
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/j.na.2010.03.004
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/j.na.2008.11.037
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: https://citeseerx.ist.psu.edu
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/j.na.2004.07.009
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/j.na.2008.03.038
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: https://dml.cz/handle/10338.dmlcz/102493
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: https://dml.cz/handle/10338.dmlcz/128252
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. http://ejde.math.unt.edu
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
Refbacks
- There are currently no refbacks.