Mathematica Bohemica, Vol. 144, No. 3, pp. 299-324, 2019


Existence of solutions for some quasilinear $\vec{p}(x)$-elliptic problem with Hardy potential

Elhoussine Azroul, Mohammed Bouziani, Hassane Hjiaj, Ahmed Youssfi

Received July 29, 2017.   Published online November 22, 2018.

Abstract:  We consider the anisotropic quasilinear elliptic Dirichlet problem $\begin{cases}-\sum_{i=1}^N D^i a_i(x,u,\nabla u) + |u|^{s(x)-1}u= f +\lambda\frac{|u|^{p_0(x)-2}u}{|x|^{p_0(x)}}&\text{in} \Omega, u = 0 & \text{on} \partial \Omega, \end{cases}$ where $\Omega$ is an open bounded subset of $\Bbb R^N$ containing the origin. We show the existence of entropy solution for this equation where the data $f$ is assumed to be in $L^1(\Omega)$ and $\lambda$ is a positive constant.
Keywords:  anisotropic variable exponent Sobolev space; quasilinear elliptic equation; Hardy potential; entropy solution; $L^1$-data
Classification MSC:  35J15, 35J62


References:
[1] B. Abdellaoui, I. Peral, A. Primo: Elliptic problems with a Hardy potential and critical growth in the gradient: non-resonance and blow-up results. J. Differ. Equations 239 (2007), 386-416. DOI 10.1016/j.jde.2007.05.010 | MR 2344278 | Zbl 1331.35128
[2] A. Alberico, G. Di Blasio, F. Feo: A priori estimates for solutions to anisotropic elliptic problems via symmetrization. Math. Nachr. 290 (2017), 986-1003. DOI 10.1002/mana.201500282 | MR 3652210 | Zbl 1375.35136
[3] S. N. Antontsev, M. Chipot: Anisotropic equations: uniqueness and existence results. Differ. Integral Equ. 21 (2008), 401-419. MR 2483260 | Zbl 1224.35088
[4] S. N. Antontsev, J. F. Rodrigues: On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 52 (2006), 19-36. DOI 10.1007/s11565-006-0002-9 | MR 2246902 | Zbl 1117.76004
[5] G. Barletta, A. Cianchi: Dirichlet problems for fully anisotropic elliptic equations. Proc. R. Soc. Edinb., Sect. A, Math. 147 (2017), 25-60. DOI 10.1017/S0308210516000020 | MR 3603525 | Zbl 1388.35043
[6] M. B. Benboubker, E. Azroul, A. Barbara: Quasilinear elliptic problems with nonstandard growth. Electron. J. Differ. Equ. 2011 (2011), Paper No. 62, 16 pages. MR 2801247 | Zbl 1221.35165
[7] M. B. Benboubker, H. Hjiaj, S. Ouaro: Entropy solutions to nonlinear elliptic anisotropic problem with variable exponent. J. Appl. Anal. Comput. 4 (2014), 245-270. MR 3226454 | Zbl 1316.35104
[8] M. Bendahmane, M. Chrif, S. El Manouni: An approximation result in generalized anisotropic Sobolev spaces and applications. Z. Anal. Anwend. 30 (2011), 341-353. DOI 10.4171/ZAA/1438 | MR 2819499 | Zbl 1231.35065
[9] M. Bendahmane, K. H. Karlsen, M. Saad: Nonlinear anisotropic elliptic and parabolic equations with variable exponents and {$L^1$} data. Commun. Pure Appl. Anal. 12 (2013), 1201-1220. DOI 10.3934/cpaa.2013.12.1201 | MR 2989682 | Zbl 1268.35053
[10] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. L. Vázquez: An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22 (1995), 241-273. MR 1354907 | Zbl 0866.35037
[11] L. Boccardo, T. Gallouët, P. Marcellini: Anisotropic equations in $L^1$. Differ. Integral Equ. 9 (1996), 209-212. MR 1364043 | Zbl 0838.35048
[12] A. Cianchi: Symmetrization in anisotropic elliptic problems. Commun. Partial Differ. Equations 32 (2007), 693-717. DOI 10.1080/03605300600634973 | MR 2334829 | Zbl 1219.35028
[13] F. C. Cirstea, J. Vétois: Fundamental solutions for anisotropic elliptic equations: existence and a priori estimates. Commun. Partial Differ. Equations 40 (2015), 727-765. DOI 10.1080/03605302.2014.969374 | MR 3299354 | Zbl 1326.35153
[14] R. Di Nardo, F. Feo: Existence and uniqueness for nonlinear anisotropic elliptic equations. Arch. Math. 102 (2014), 141-153. DOI 10.1007/s00013-014-0611-y | MR 3169023 | Zbl 1293.35108
[15] R. Di Nardo, F. Feo, O. Guibé: Uniqueness result for nonlinear anisotropic elliptic equations. Adv. Differ. Equ. 18 (2013), 433-458. MR 3086461 | Zbl 1272.35092
[16] L. Diening, P. Harjulehto, P. Hästö, R. M. Růžička: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017. Springer, Berlin (2011). DOI 10.1007/978-3-642-18363-8 | MR 2790542 | Zbl 1222.46002
[17] R. J. DiPerna, P.-L. Lions: On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. (2) 130 (1989), 321-366. DOI 10.2307/1971423 | MR 1014927 | Zbl 0698.45010
[18] R. J. DiPerna, P.-L. Lions: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511-547. DOI 10.1007/BF01393835 | MR 1022305 | Zbl 0696.34049
[19] O. Guibé: Uniqueness of the renormalized solution to a class of nonlinear elliptic equations. On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments (A. Alvino et al., eds.). Quad. Mat. 23. Dipartimento di Matematica, Seconda Università di Napoli, Caserta; Aracne, Rome. (2008), 256-282. MR 2762168 | Zbl 1216.35036
[20] O. Guibé, A. Mercaldo: Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data. Trans. Am. Math. Soc. 360 (2008), 643-669. DOI 10.1090/S0002-9947-07-04139-6 | MR 2346466 | Zbl 1156.35042
[21] P. Gwiazda, I. Skrzypczak, A. Zatorska-Goldstein: Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space. J. Differ. Equations 264 (2018), 341-377. DOI 10.1016/j.jde.2017.09.007 | MR 3712945 | Zbl 1376.35046
[22] J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Etudes mathematiques. Dunod; Gauthier-Villars, Paris (1969) (In French.). MR 0259693 | Zbl 0189.40603
[23] Y. Liu, R. Davidson, P. Taylor: Investigation of the touch sensitivity of ER fluid based tactile display. Smart Structures and Materials: Smart Structures and Integrated Systems. Proceeding of SPIE 5764 (2005), 92-99. DOI 10.1117/12.598713
[24] M. Mihăilescu, P. Pucci, V. Rădulescu: Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent. J. Math. Anal. Appl. 340 (2008), 687-698. DOI 10.1016/j.jmaa.2007.09.015 | MR 2376189 | Zbl 1135.35058
[25] F. Mokhtari: Regularity of the solution to nonlinear anisotropic elliptic equations with variable exponents and irregular data. Mediterr. J. Math. 14 (2017), Article No. 141, 18 pages. DOI 10.1007/s00009-017-0941-7 | MR 3656509 | Zbl 1377.35102
[26] M. M. Porzio: On some quasilinear elliptic equations involving Hardy potential. Rend. Mat. Appl., VII. Ser. 27 (2007), 277-297. MR 2398427 | Zbl 1156.35044
[27] J. Vétois: Existence and regularity for critical anisotropic equations with critical directions. Adv. Differ. Equ. 16 (2011), 61-83. MR 2766894 | Zbl 1220.35081
[28] J. Vétois: Strong maximum principles for anisotropic elliptic and parabolic equations. Adv. Nonlinear Stud. 12 (2012), 101-114. DOI 10.1515/ans-2012-0106 | MR 2895946 | Zbl 1247.35007
[29] P. Wittbold, A. Zimmermann: Existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponents and {$L^1$}-data. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 2990-3008. DOI 10.1016/j.na.2009.11.041 | MR 2580154 | Zbl 1185.35088
[30] A. Youssfi, E. Azroul, H. Hjiaj: On nonlinear elliptic equations with Hardy potential and {$L^1$}-data. Monatsh. Math. 173 (2014), 107-129. DOI 10.1007/s00605-013-0516-z | MR 3148663 | Zbl 1285.35035

Affiliations:   Elhoussine Azroul, Mohammed Bouziani, Department of Mathematics, Laboratory LAMA, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, BP 1796 Atlas, Fez, Morocco, e-mail: elhoussine.azroul@gmail.com, mhdbouziani2010@gmail.com; Hassane Hjiaj, Department of Mathematics, Faculty of Sciences, Tetouan University Abdelmalek Essaadi, Quartier M'haneche II, Avenue Palestine, BP 2121, Tetouan 93000, Morocco, e-mail: hjiajhassane@yahoo.fr; Ahmed Youssfi, National School of Applied Sciences, Laboratory of Engineering, Systems and Applications (LISA), Sidi Mohamed Ben Abdellah University, My Abdellah Avenue, Imouzzer road, P.O. Box 72 Fs-Principale, 30 000 Fez, Morocco, e-mail: ahmed.youssfi@gmail.com


 
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