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Regularity for solutions of nonlinear elliptic equations. (English) Zbl 0842.35014
The authors study Dirichlet problems for instance of type $-\text{div} \bigl( a(x,u, Du) \bigr) + d(x) Du |Du |^{p - 1} + H(x,u) = \text{div} f \text{ in } \Omega, \;u \in W^{1,p}_0 (\Omega)$ under structural assumptions on $$a,d,H$$ and $$f$$, which allow for a comparison of $$u$$ via Schwarz-symmetrization with a solution $$v(r)$$ of the associated problem $$-\Delta_p v = \text{div} F(r)$$ in $$\Omega^\#$$, $$u \in W_0^{1,p} (\Omega^\#)$$. The construction of $$F$$ from the data is quite interesting and nonstandard. Regularity of $$u$$ in Orlicz spaces is derived when $$|f|$$ and coefficients of the equation belong to suitable Lorentz spaces.
Reviewer: B.Kawohl (Köln)

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35B45 A priori estimates in context of PDEs 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems