On the existence solutions for some Nonlinear elliptic problem
Abstract
In the present paper, we study the existence and regularity of positive solutions for the following boundary value problem : $\mathrm{-div}\> \big( \lvert\nabla u\rvert^{p-2}\nabla u ) + u^{s} = \dfrac{f}{u^{\alpha}}\mbox{ in } \Omega \mbox{ and } u=0\mbox{ on } \partial\Omega,$ where $ \Omega $ is an open and bounded subset of $ \mathbb{R}^{N} $ $ (N> p>1) $, $ 0<\alpha\leq 1 $, $ s\geq 1 $ and $f$ is a nonnegative function that belongs to some Lebesgue space.
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Akdim. Y, Benkirane. A and El Moumni. M, Solutions of nonlinear elliptic problems withlower order terms, Ann. Funct. Anal. 6, no. 1, 34-53 (2015). https://doi.org/10.15352/afa/06-1-4
Ambrosetti. A, Brezis. H and Cerami. G, Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122(2), 519-543 (1994). https://doi.org/10.1006/jfan.1994.1078
Benkirane. A, El Haji. B and El Moumni. M, On the existence of solution for degenerate parabolic equations with singular terms. Pure and Applied Mathematics Quarterly Volume 14, Number 3-4, (2018), 591-606 https://doi.org/10.4310/PAMQ.2018.v14.n3.a8
Boccardo. L, Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data, Ann Mat. Pura Appl. (4 ) 188, no. 4, pp. 591-601 (2009). https://doi.org/10.1007/s10231-008-0090-5
Boccardo. L and F. Murat. F, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Analysis 19 (1992) 581-597. https://doi.org/10.1016/0362-546X(92)90023-8
Boccardo. L, Gallouet. T, Vazquez. J. L, Nonlinear elliptic equations in RN without growth restrictions on the data, J. Differential Equations 105, no. 2, pp. 334-363 (1993). https://doi.org/10.1006/jdeq.1993.1092
Boccardo. L and Orsina. L, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37, pp. 363-380 (2009). https://doi.org/10.1007/s00526-009-0266-x
Callegari. A and Nashman. A, A nonlinear singular boundary-value problem in the theory of pseudoplastic fluids. SIAM J. Appl. Math. 38(2), 275-281 (1980). https://doi.org/10.1137/0138024
Canino. A, Sciunzi. B and Trombetta. A, Existence and uniqueness for p-Laplace equations involving singular nonlinearities, NoDEA Nonlinear Differential Equations Appl. 23, pp. 8-18 (2016). https://doi.org/10.1007/s00030-016-0361-6
Cirmi. R, Regularity of the solutions to nonlinear elliptic equations with a lower-order term, Nonlinear Anal. T.M.A. 25, pp. 569-580 (1995). https://doi.org/10.1016/0362-546X(94)00173-F
Chu. Y, Gao. R and Sun. y, Existence and regularity of solutions to a quasilinear elliptic problem involving variable sources, Boundary Value Problems 2017:155. https://doi.org/10.1186/s13661-017-0888-4
Dall'Aglio. A, Orsina. L and Petitta. F, Existence of solutions for degenerate parabolic equations with singular terms. Nonlinear Anal. 131, 273-288 (2016). https://doi.org/10.1016/j.na.2015.06.030
De Cave. L, Nonlinear elliptic equations with singular nonlinearities, Asymptot. Anal. 84 (3-4), pp. 181-195(2013). https://doi.org/10.3233/ASY-131173
De Cave. L and F. Oliva. F, On the regularizing effet of some absorption and singular lower order terms in classical Dirichlet problems with L1 data, JEPE. Vol 2, p. 73-85 (2016). https://doi.org/10.1007/BF03377393
El Haji. B, El Moumni. M and Kouhaila. K, On a nonlinear elliptic problems having large monotonocity with L1 data in weighted Orlicz-Sobolev spaces. Moroccan J. of Pure and Appl. Anal. (MJPAA) Vol 5(1), (2019), 104-116. https://doi.org/10.2478/mjpaa-2019-0008
El Moumni. M, Nonlinear Elliptic Equations Without Sign Condition and L1 -Data in Musielak-Orlicz-Sobolev Spaces, Acta Applicandae Mathematicae https://doi.org/10.1007/s10440-018-0186-x
Keller. H. B and Cohen. D. S, Some positive problems suggested by nonlinear heat generators. J. Math. Mech. 16(12), 1361-1376 (1967). https://doi.org/10.1512/iumj.1967.16.16087
Leray. J and Lions. J- L, Quelques resultats de Visik sur les problemes elliptiques non lineaires par les methodes de Minty-Browder, Bull. Soc. Math. France, 93, pp. 97-107 (1965). https://doi.org/10.24033/bsmf.1617
Maso. G. D, Murat. F, Orsina. L and Prignet. A , Renormalized solutions of elliptic equations with general measure data. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 28(4), 741-808 (1999).
Oliva. F and Petitta. F, Finite and infinite energy solutions of singular elliptic problems: existence and uniqueness. J. Differential Equations 264, no. 1, 311-340 (2018). https://doi.org/10.1016/j.jde.2017.09.008
Oliva. F and Petitta. F, On singular elliptic equations with measure sources. ESAIM Control Optim. Calc. Var. 22, no. 1, 289-308 (2016). https://doi.org/10.1051/cocv/2015004
Stampacchia. G, Equations elliptiques du second ordre a coeficients discontinus, Les Presses de l'Universite de Montreal 1966.
Vazquez. J. L, A Strong Maximum Principle for Some Quasilinear Elliptic Equations, Appl. Math. Optim., 12, pp. 191-202 (1984). https://doi.org/10.1007/BF01449041
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