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A Singular Quasilinear Anisotropic Elliptic Boundary Value Problem. II. (English) Zbl 0901.35031

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.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
35B45 A priori estimates in context of PDEs

Citations:

Zbl 0835.35049
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Full Text: DOI

References:

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