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Zbl 1171.35057
Pucci, Patrizia; Servadei, Raffaella
Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations.
(English)
[J] Indiana Univ. Math. J. 57, No. 7, 3329-3364 (2008). ISSN 0022-2518

The authors consider elliptic problems of the form $\nabla\cdot{\bold A}(x,u,\nabla u)=B(x,u,\nabla u)$ in $\Omega$, where $\Omega\subseteq{\Bbb R}^n$ is not necessary a bounded domain. The principal part can degenerate, e.g., it is a $p$-Laplacian with $1<p<n$, or in the case inhomogeneous $A(x,\xi)= \left\vert\xi\right\vert^{p-2} \xi \Bigl(1-\log\bigl( {{1+\left\vert\xi\right\vert}\over{\left\vert\xi\right\vert}}\bigr) \Bigr)$ for $\xi\in{\Bbb R}^n\setminus\{0\}$. They obtain conditions for weak solutions $u\in W^{1,p}(\Omega)$ to belong to $L^m_{ loc}(\Omega)$, $1\le m\le \infty$, and to $W^{2,p}_{ loc}(\Omega)$. They also deal with radial weak solutions. The proofs are based on the Moser iteration scheme and Nirenberg's translation method. Further results on the radial case appeared in [{\it P. Pucci} and {\it R. Servadei}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, No.~3, 505--537 (2008; Zbl 1147.35045)].
[Georg Hetzer (Auburn)]
MSC 2000:
*35J70 Elliptic equations of degenerate type
35J60 Nonlinear elliptic equations
35D10 Regularity of generalized solutions of PDE

Keywords: quasilinear elliptic equations; degenerate type; weak solutions; regularity

Citations: Zbl 1147.35045

Cited in: Zbl 1223.35128

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