Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
anisotropic Sobolev space; weak solution; entropy solution; strongly nonlinear problem; Neumann boundary condition; unilateral problem
anisotropic Sobolev space; weak solution; entropy solution; strongly nonlinear problem; Neumann boundary condition; unilateral problem
Summary:
We consider the following strongly nonlinear Neumann elliptic problem: $$\begin {cases} \displaystyle -\sum ^{N}_{i=1} D^{i} a_{i}(x,u,\nabla u) + H(x,u,\nabla u)+|u|^{p_{0}-2}u = f(x)+\sum ^{N}_{i=1} D^{i} \phi _{i}(x,u) & \mbox {in} \ \Omega ,\\ \displaystyle \sum _{i=1}^{N}(a_{i}(x,u,\nabla u) - \phi _{i}(x,u))\cdot n_{i}= 0 & \mbox {on} \ \partial \Omega , \end {cases}$$ where the Carathéodory functions $a_{i}(x,s,\xi ),$ $H(x,s,\xi )$ and $\phi _{i}(x,s)$ verify some nonstandard conditions. By applying an approximation method, we prove the existence of entropy solutions for the unilateral problem with $L^{1}$-data, and we conclude some regularity results.
References:
[1] Akdim, A., Azroul, E., Benkirane, A.: Existence of solutions for quasilinear degenerate elliptic equations. Electron. J. Differ. Equ. 2001 (2001), Article ID 71, 19 pages. MR 1872050 | Zbl 0988.35065
[2] Akdim, Y., Belayachi, M., Hjiaj, H., Mekkour, M.: Entropy solutions for some nonlinear and noncoercive unilateral elliptic problems. Rend. Circ. Mat. Palermo (2) 69 (2020), 1373-1392. DOI 10.1007/s12215-019-00477-2 | MR 4168179 | Zbl 1464.35108
[3] Alvino, A., Boccardo, L., Ferone, V., Orsina, L., Trombetti, G.: Existence results for nonlinear elliptic equations with degenerate coercivity. Ann. Mat. Pura Appl., IV. Ser. 182 (2003), 53-79. DOI 10.1007/s10231-002-0056-y | MR 1970464 | Zbl 1105.35040
[4] Andreu, F., León, J. M. Mazón S. Segura de, Toledo, J.: Quasi-linear elliptic and parabolic equations in $L^1$ with nonlinear boundary conditions. Adv. Math. Sci. Appl. 7 (1997), 183-213. MR 1454663 | Zbl 0882.35048
[5] Antontsev, S., Chipot, M.: Anisotropic equations: Uniqueness and existence results. Differ. Integral Equ. 21 (2008), 401-419. DOI 10.57262/die/1356038624 | MR 2483260 | Zbl 1224.35088
[6] Benaichouche, N., Ayadi, H., Mokhtari, F., Hakem, A.: Existence and regularity results for nonlinear anisotropic unilateral elliptic problems with degenerate coercivity. J. Elliptic Parabol. Equ. 8 (2022), 171-195. DOI 10.1007/s41808-022-00148-x | MR 4426333 | Zbl 1491.35235
[7] Benboubker, M. B., Benkhalou, H., Hjiaj, H.: Weak and renormalized solutions for anisotropic Neumann problems with degenerate coercivity. Bol. Soc. Parana. Mat. (3) 41 (2023), Article ID 144, 25 pages. DOI 10.5269/bspm.62362 | MR 4541250 | Zbl 07805702
[8] Benboubker, M. B., Chrayteh, H., Hjiaj, H., Yazough, C.: Existence of solutions in the sense of distributions of anisotropic nonlinear elliptic equations with variable exponent. Topol. Methods Nonlinear Anal. 46 (2015), 665-693. DOI 10.12775/TMNA.2015.063 | MR 3494963 | Zbl 1362.35130
[9] Bendahmane, M., Chrif, M., Manouni, S. El: 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
[10] Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J. L.: 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] Boccardo, L., Dall'Aglio, A., Orsina, L.: Existence and regularity results for some elliptic equations with degenerate coercivity. Atti Semin. Mat. Fis. Univ. Modena 46 Suppl. (1998), 51-81. MR 1645710 | Zbl 0911.35049
[12] Boccardo, L., Gallouët, T., Marcellini, P.: Anisotropic equations in $L^1$. Differ. Integral Equ. 9 (1996), 209-212. DOI 10.57262/die/1367969997 | MR 1364043 | Zbl 0838.35048
[13] Dakkak, A., Hjiaj, H., Sanhaji, A.: Existence of $T-\vec{p}(.)$-solutions for some quasilinear anisotropic elliptic problem. Rend. Mat. Appl., VII. Ser. 40 (2019), 113-140. MR 4026087 | Zbl 1440.35063
[14] Nardo, R. Di, Feo, F.: 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] Nardo, R. Di, Feo, F., Guibé, O.: Uniqueness result for nonlinear anisotropic elliptic equations. Adv. Differ. Equ. 18 (2013), 433-458. DOI 10.57262/ade/1363266253 | MR 3086461 | Zbl 1272.35092
[16] Fan, X.: Anisotropic variable exponent Sobolev spaces and $\vec{p}(x)$-Laplacian equations. Complex Var. Elliptic Equ. 56 (2011), 623-642. DOI 10.1080/17476931003728412 | MR 2832206 | Zbl 1236.46029
[17] Li, F.: Anisotropic elliptic equations in $L^m$. J. Convex. Anal. 8 (2001), 417-422. MR 1915951 | Zbl 1004.35052
[18] Mihăilescu, M., Pucci, P., Rădulescu, V.: 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
[19] Salmani, A., Akdim, Y., Redwane, H.: Entropy solutions of anisotropic elliptic nonlinear obstacle problem with measure data. Ric. Mat. 69 (2020), 121-151. DOI 10.1007/s11587-019-00452-0 | MR 4098176 | Zbl 1440.35117
[20] Troisi, M.: Teoremi di inclusione per spazi di Sobolev non isotropi. Ric. Mat. 18 (1969), 3-24 Italian. MR 0415302 | Zbl 0182.16802
Search
Partner of
