Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations. (English) Zbl 1171.35057

The authors consider elliptic problems of the form \(\nabla\cdot{\mathbf A}(x,u,\nabla u)=B(x,u,\nabla u)\) in \(\Omega\), where \(\Omega\subseteq{\mathbb R}^n\) 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 \(A(x,\xi)= \left|\xi\right|^{p-2} \xi \Bigl(1-\log\bigl( {{1+\left|\xi\right|}\over{\left|\xi\right|}}\bigr) \Bigr)\) for \(\xi\in{\mathbb R}^n\setminus\{0\}\). They obtain conditions for weak solutions \(u\in W^{1,p}(\Omega)\) to belong to \(L^m_{ loc}(\Omega)\), \(1\leq m\leq \infty\), 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 [P. Pucci and R. Servadei, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, No. 3, 505–537 (2008; Zbl 1147.35045)].


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


Zbl 1147.35045
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