Choi, Y. S.; McKenna, P. J. A Singular Quasilinear Anisotropic Elliptic Boundary Value Problem. II. (English) Zbl 0901.35031 Trans. Am. Math. Soc. 350, No. 7, 2925-2937 (1998). Summary: [For part I of this paper, see Y. S. Choi, A. C. Lazer and P. J. McKenna, Trans. Am. Math. Soc. 347, 2633-2641 (1995; Zbl 0835.35049).]Let \(\Omega \subset \mathbb{R}^N\) with \(N \geq 2\). We consider the equations \[ \sum_{i=1}^{N} u^{a_i} \frac{\partial^2 u}{\partial x_i^2} +p(x)= 0, \qquad u| _{\partial\Omega}= 0, \] with \(a_1 \geq a_2 \geq\dots \geq a_N \geq 0\) and \(a_1>a_N\). We show that if \(\Omega\) is a convex bounded region in \(\mathbb{R}^N\), there exists at least one classical solution to this boundary value problem. If the region is not convex, we show the existence of a weak solution. Partial results for the existence of classical solutions for non-convex domains in \(\mathbb{R}^2\) are also given. Cited in 16 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35J70 Degenerate elliptic equations 35B45 A priori estimates in context of PDEs Keywords:Harnack inequality; subsolution; supersolution Citations:Zbl 0835.35049 PDFBibTeX XMLCite \textit{Y. S. Choi} and \textit{P. J. McKenna}, Trans. Am. Math. Soc. 350, No. 7, 2925--2937 (1998; Zbl 0901.35031) Full Text: DOI References: [1] Sunčica Čanić and Barbara Lee Keyfitz, An elliptic problem arising from the unsteady transonic small disturbance equation, J. Differential Equations 125 (1996), no. 2, 548 – 574. · Zbl 0869.35043 · doi:10.1006/jdeq.1996.0040 [2] Y. S. Choi, A. C. Lazer, and P. J. McKenna, On a singular quasilinear anisotropic elliptic boundary value problem, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2633 – 2641. · Zbl 0835.35049 [3] M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), no. 2, 193 – 222. · Zbl 0362.35031 · doi:10.1080/03605307708820029 [4] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001 [5] A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc. 111 (1991), no. 3, 721 – 730. · Zbl 0727.35057 [6] A. Nachman and A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math. 38 (1980), no. 2, 275 – 281. · Zbl 0453.76002 · doi:10.1137/0138024 [7] W. Reichel, Uniqueness for degenerate elliptic equations via Serrin’s sweeping principle, General Inequalities 7, International Series of Numerical Mathematics, Birkhäuser, Basel, 1997, pp. 375-387. CMP 97:14 [8] David H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, Vol. 309, Springer-Verlag, Berlin-New York, 1973. · Zbl 0248.35003 [9] C. A. Stuart, Existence theorems for a class of non-linear integral equations, Math. Z. 137 (1974), 49 – 66. · Zbl 0289.45013 · doi:10.1007/BF01213934 [10] Steven D. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal. 3 (1979), no. 6, 897 – 904. · Zbl 0421.34021 · doi:10.1016/0362-546X(79)90057-9 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.