ON SOLVABILITY OF A CLASS OF NONLINEAR ELLIPTIC TYPE EQUATION WITH VARIABLE EXPONENT

Uğur Sert; Kamal Soltanov

doi:10.11948/2017071

2017 Volume 7 Issue 3

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Uğur Sert, Kamal Soltanov. ON SOLVABILITY OF A CLASS OF NONLINEAR ELLIPTIC TYPE EQUATION WITH VARIABLE EXPONENT[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 1139-1160. doi: 10.11948/2017071

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Uğur Sert, Kamal Soltanov. ON SOLVABILITY OF A CLASS OF NONLINEAR ELLIPTIC TYPE EQUATION WITH VARIABLE EXPONENT[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 1139-1160. doi: 10.11948/2017071

ON SOLVABILITY OF A CLASS OF NONLINEAR ELLIPTIC TYPE EQUATION WITH VARIABLE EXPONENT

Department of Mathematics, Hacettepe Uni., 06800, Beytepe, Ankara, Turkey

Abstract

Abstract

In this paper, we study the Dirichlet problem for the implicit degenerate nonlinear elliptic equation with variable exponent in a bounded domain Ω ⊂ Rⁿ. We obtain sufficient conditions for the existence of a solution without regularization and any restriction between the exponents. Furthermore, we define the domain of the operator generated by posed problem and investigate its some properties and also its relations with known spaces that enable us to prove existence theorem.
- PDEs with nonstandart nonlinearity /
- solvability theorem /
- variable exponent /
- implicit degenerate PDEs
MSC: 35J60;35J66

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References

[1]	E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological liquids, Arch. Ration. Mech. Anal., 2002, 164, 213-259. Google Scholar
[2]	T. Adamowicz and O. Toivanen, Hölder continuity of quasiminimizers with nonstandard growth, Nonlinear Anal., 2015, 125, 433-456. Google Scholar
[3]	R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar
[4]	S. N. Antontsev and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity:existence, uniqueness and localization properties of solutions, Nonlinear Anal., 2005, 60, 515-545. Google Scholar
[5]	S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara, Sez. VⅡ, Sci. Mat., 2006, 52(1), 19-36. Google Scholar
[6]	S. N. Antontsev and S. I. Shmarev, On the localization of solutions of elliptic equations with nonhomogeneous anisotropic degeneration, Siberian Math. J., 2005, 5, 765-782. Google Scholar
[7]	S. N. Antontsev and S. I. Shmarev, Elliptic equations and systems with nonstandard growth conditions:Existence, uniqueness and localization properties of solutions, Nonlinear Anal., 2006, 65, 728-761. Google Scholar
[8]	Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 2006, 66, 1383-1406. Google Scholar
[9]	L. Diening, Theoretical and numerical results for electrorheological fluids, Ph.D. Thesis, 2002. Google Scholar
[10]	Yu. A. Dubinskii, Weak convergence in nonlinear elliptic and parabolic equations, (Russian)Mat.Sb., 1965, 67(109), 609-642. Google Scholar
[11]	X. Fan and D. Zhao, On the spaces L^p(x)(Ω) and W^k,p(x)(Ω), J. Math. Anal. Appl., 2001, 263, 424-446. Google Scholar
[12]	X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces W^k,p(x), J. Math. Anal. Appl., 2001, 262(2), 749-760. Google Scholar
[13]	X. Fan and D. Zhao, The quasi-minimizer of integral functionals with m(x) growth conditions, Nonlinear Anal., 2000, 39(7), 807-816. Google Scholar
[14]	Y. Fu and Y. Shan, On the removability of isolated singular points for elliptic equations involving variable exponent, Adv. Nonlinear Anal., 2016, 5(2), 121-132. Google Scholar
[15]	S. Fucik and A. Kufner, Nonlinear Differential Equations, Elsevier, New York, 1980. Google Scholar
[16]	M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta. Math., 1987, 59, 245-248. Google Scholar
[17]	H. Hudzik, On generalized Orlicz-Sobolev space, Funct. Approx. Comment. Math., 1976, 4, 37-51. Google Scholar
[18]	O. Kovacik and J. Rakosnik, On spaces L^p(x) and W^k,p(x), Czechoslovak Math. J., 1991, 41, 592-618. Google Scholar
[19]	J. L. Lions, Queques Methodes de Resolution des Problemes Aux Limites Non lineaires, Dunod and Gauthier-Villars, Paris, 1969. Google Scholar
[20]	P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rational Mech. Anal., 1989, 105, 267-284. Google Scholar
[21]	G. de Marsily, Quantitative Hydrogeology. Groundwater Hydrology for Engineers, Academic Press, London, 1986. Google Scholar
[22]	J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983. Google Scholar
[23]	V. Rădulescu and D. Repovš, Partial Differential Equations with Variable Exponents:Variational methods and Quantitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015. Google Scholar
[24]	V. Rădulescu, Nonlinear elliptic equations with variable exponent:old and new, Nonlinear Anal., 2015, 121, 336-369. Google Scholar
[25]	D. Repovš, Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. (Singap.), 2015, 13(6), 645-661. Google Scholar
[26]	M. M. Rao, Measure Theory and Integration, John Wiley & Sons, New York, 1984. Google Scholar
[27]	P. A. Raviart, Sur la resolution et l'approximation de certaines equations paraboliques non lineaires, Arch. Rational Mech. Anal., 1967, 25, 64-80. Google Scholar
[28]	M. Ruzicka, Electrorheological Fluids:Modeling and Mathematical Theory, In Lecture Notes in Mathematics, Springer, Berlin, 2000. Google Scholar
[29]	K. N. Soltanov and J. Sprekels, Nonlinear equations in non-reflexive Banach spaces and strongly nonlinear equations, Adv. Math. Sci. Appl., 1999, 9(2), 939-972. Google Scholar
[30]	K. N. Soltanov and M. A. Ahmadov, On nonlinear parabolic equation in nondivergent form with implicit degeneration and embedding theorems, 2012, arXiv:1207.7063v1[math.AP]. Google Scholar
[31]	K. N. Soltanov, Some imbedding theorems and nonlinear differential equations, Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 1999, 19(5), Math. Mech., 2000, 125-146, (Reviewer:H. Triebel). Google Scholar
[32]	K. N. Soltanov, Some nonlinear equations of the nonstable filtration type and embedding theorems, Nonlinear Anal., 2006, 65, 2103-2134. Google Scholar
[33]	K. N. Soltanov, On noncoercive semilinear equations, Nonlinear Anal. Hybrid Syst., 2008, 2, 344-358. Google Scholar
[34]	K. N. Soltanov, Some Applications of Nonlinear Analysis to Differential Equations, (in Russian)ELM, Baku, 2002. Google Scholar
[35]	K. N. Soltanov, Some embedding theorems and its applications to nonlinear equations, Differensial'nie Uravnenia., 1984, 20(12), 2181-2184. Google Scholar
[36]	K. N. Soltanov, On some modification Navier-Stokes equations, Nonlinear Anal., 2003, 52(3), 769-793. Google Scholar
[37]	Z. Yücedağ, Solutions of nonlinear problems involving p(x)-Laplacian operator, Adv. Nonlinear Anal., 2015, 4, 285-293. Google Scholar
[38]	V. V. Zhikov, On some variational problems, Russian J. Math. Phys., 1997, 5(1), 105-116. Google Scholar
[39]	V. V. Zhikov, On the technique for passing to the limit in nonlinear elliptic equations, Functional Anal. and Its App., 2009, 43(2), 96-112. Google Scholar
[40]	V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv., 1987, 29, 33-36. Google Scholar