Local behavior of solutions of quasilinear elliptic equations with general structure. (English) Zbl 0708.35023

Rakotoson, J. M.; Ziemer, William P.

Trans. Am. Math. Soc. 319, No. 2, 747-775 (1990).

The authors discuss weak solutions of quasilinear equations of the type \[ -div A(x,u,\nabla u)+B(x,u,\nabla u)=T \] on a domain \(\Omega \subset {\mathbb{R}}^ n\) where A,B satisfy natural structure and growth conditions, e.g. \[ A(x,\eta,\xi)\cdot \xi \geq v_ 0(x)| \xi |^ p-v_ 1(x),\quad | A(x,\eta,\xi)| \leq c_ 0(x)| \xi |^{p- 1}+a_ 0(x) \] for some exponent \(1<p<n\). T is a distribution belonging to some space \(W^{-1,q}(\Omega)\) but the authors mostly concentrate on the case when T is represented by a nonnegative Radon-measure \(\mu\) and show that a growth condition of the form \(\mu (B_ r(x))\leq M\cdot r^{n-p+\epsilon}\) for all balls \(B_ r(x)\subset {\mathbb{R}}^ n\) is equivalent to the Hölder-continuity of weak solutions.

Reviewer: M.Fuchs

MSC:

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

Keywords:

Radon-measure; growth condition; Hölder-continuity

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