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Transactions of the American Mathematical Society

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A Singular Quasilinear Anisotropic Elliptic Boundary Value Problem. II


Authors: Y. S. Choi and P. J. McKenna
Journal: Trans. Amer. Math. Soc. 350 (1998), 2925-2937
MSC (1991): Primary 35J25
DOI: https://doi.org/10.1090/S0002-9947-98-02276-4
MathSciNet review: 1491858
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Abstract: Let $\Omega \subset \mathbf{R}^N$ with $N \geq 2$. We consider the equations

\begin{displaymath}\begin{array}{rcl} \displaystyle \sum _{i=1}^{N} u^{a_i} \frac{\partial^2 u}{\partial x_i^2} +p(\mathbf{x})& = & 0, \\ u|_{\partial\Omega} & = & 0, \end{array} \end{displaymath}

with $a_1 \geq a_2 \geq .... \geq a_N \geq 0$ and $a_1>a_N$. We show that if $\Omega$ is a convex bounded region in $\mathbf{R}^N$, 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 $\mathbf{R}^2$ are also given.


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Additional Information

Y. S. Choi
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06268
Email: choi@math.uconn.edu

P. J. McKenna
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06268
Email: mckenna@math.uconn.edu

DOI: https://doi.org/10.1090/S0002-9947-98-02276-4
Keywords: Harnack inequality, singular, subsolution, supersolution
Received by editor(s): August 6, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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