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Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations. (English) Zbl 1171.35057
The authors consider elliptic problems of the form $\nabla ·𝐀\left(x,u,\nabla u\right)=B\left(x,u,\nabla u\right)$ in ${\Omega }$, where ${\Omega }\subseteq {ℝ}^{n}$ is not necessary a bounded domain. The principal part can degenerate, e.g., it is a $p$-Laplacian with $1, or in the case inhomogeneous $A\left(x,\xi \right)={\left|\xi \right|}^{p-2}\xi \left(1-log\left(\frac{1+\left|\xi \right|}{\left|\xi \right|}\right)\right)$ for $\xi \in {ℝ}^{n}\setminus \left\{0\right\}$. They obtain conditions for weak solutions $u\in {W}^{1,p}\left({\Omega }\right)$ to belong to ${L}_{loc}^{m}\left({\Omega }\right)$, $1\le m\le \infty$, and to ${W}_{loc}^{2,p}\left({\Omega }\right)$. 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 [P. Pucci and R. Servadei, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, No. 3, 505–537 (2008; Zbl 1147.35045)].

##### MSC:
 35J70 Degenerate elliptic equations 35J60 Nonlinear elliptic equations 35D10 Regularity of generalized solutions of PDE (MSC2000)