Aouaoui, Sami Quasilinear equations with degenerate coerciveness and having multiple singularities. (English) Zbl 1307.35113 Complex Var. Elliptic Equ. 59, No. 12, 1650-1663 (2014). The purpose of the present paper is on giving sufficient conditions for having both a positive bounded solution and a distributional solution (Definition 1.1) of the following differential equation (for \(x\in \mathbb R^N\)) \[ \begin{aligned} -\text{div}\left(\frac{|\nabla u(x)|^{p-2}\nabla u(x)}{(u(x))^k}\mathbf{1}_{u^{-1}((0,\infty))}\right)+|u(x)|^{p-2}u(x)+\Lambda (x,u(x))\mathbf{1}_{u^{-1}((0,\infty))}|\nabla u(x)|^{p}\\ =f(x,u(x))+h(x),\end{aligned} \] these conditions are bearing on the positive bounded integrable function \(h\) and on \(f,\Lambda\) (which presents a singular behaviour at the origin), the real-valued Carathéodory on \(\mathbb R^N\times (0,\infty)\) and on \(\mathbb R^N\times\mathbb R\), respectively, that is, measurables w.r.t. the first component and continous w.r.t. the second one (Theorems 1.1 and 1.2). \(\mathbf{1}_{u^{-1}((0,\infty))}\) stands for the indicator function on the open set \(u^{-1}((0,\infty))\) in \(\mathbb R^N\). Reviewer: Mohammed El Aïdi (Bogotá) MSC: 35J62 Quasilinear elliptic equations 35D30 Weak solutions to PDEs 35J70 Degenerate elliptic equations 35J75 Singular elliptic equations Keywords:multiple singularities; degenerate coerciveness; approximation; a priori estimates PDFBibTeX XMLCite \textit{S. Aouaoui}, Complex Var. Elliptic Equ. 59, No. 12, 1650--1663 (2014; Zbl 1307.35113) Full Text: DOI References: [1] DOI: 10.1016/j.jmaa.2008.09.073 · Zbl 1161.35013 · doi:10.1016/j.jmaa.2008.09.073 [2] DOI: 10.1016/j.jde.2009.01.016 · Zbl 1173.35051 · doi:10.1016/j.jde.2009.01.016 [3] DOI: 10.1051/cocv:2008031 · Zbl 1147.35034 · doi:10.1051/cocv:2008031 [4] Giachetti D, Boll. Un. Mat. Ital. B 2 pp 349– (2009) [5] Orsina L, Adv. Calc. Var 4 pp 397– (2011) [6] DOI: 10.1016/j.jde.2007.01.007 · Zbl 1131.35023 · doi:10.1016/j.jde.2007.01.007 [7] Zhou W, Commun. Math. Anal 5 pp 76– (2008) [8] DOI: 10.1002/mma.996 · Zbl 1155.35379 · doi:10.1002/mma.996 [9] Leray J, Bull. Soc. Math. France 93 pp 97– (1965) [10] DOI: 10.1051/cocv:2008072 · Zbl 1189.35109 · doi:10.1051/cocv:2008072 [11] DOI: 10.1016/S0021-7824(01)01211-9 · Zbl 1134.35358 · doi:10.1016/S0021-7824(01)01211-9 [12] DOI: 10.1016/j.anihpc.2005.02.006 · Zbl 1103.35040 · doi:10.1016/j.anihpc.2005.02.006 [13] DOI: 10.1512/iumj.2009.58.3409 · Zbl 1171.35011 · doi:10.1512/iumj.2009.58.3409 [14] Boccardo L, Ann. Sc. Norm. Super. Pisa Cl. Sci. tome 11 pp 213– (1984) [15] Antontsev S, Handbook of differential equations, stationary partial differential equations, Vol. 3, Chap. 1 (2006) [16] DOI: 10.1080/03605300701518208 · Zbl 1147.35038 · doi:10.1080/03605300701518208 [17] DOI: 10.1016/S0362-546X(98)00057-1 · Zbl 0930.35074 · doi:10.1016/S0362-546X(98)00057-1 [18] DOI: 10.1007/BF01449041 · Zbl 0561.35003 · doi:10.1007/BF01449041 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.