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Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. (English) Zbl 0803.35046
This paper is devoted to the study of renormalized solutions of $-\text{div} \bigl( a(x,u,Du) \bigr) - \text{div} \bigl( \Phi (u) \bigr) + g(x,u) = f, \tag{*}$ on a bounded open subset of $$\mathbb{R}^ N$$ with zero boundary conditions. The operator $$A$$ defined on $$W^{1,p} (\Omega)$$ (with $$p>1)$$ by $$(a$$ is a vector valued Caratheodory function) $$Au = \text{div} (a(x,u,Du))$$ is a nonlinear Leray-Lions operator such that there exist $$c_ i>0$$ $$(i=1,2,3)$$ such that $\bigl | a(x,t, \xi) \bigr | \leq c_ 1 | t |^{p-1} + c_ 1 | \xi |^{p-1} + d(x), \quad d \in L^{p/(p-1)} (\Omega)$ $\bigl[ a(x,t,\xi) - a(x,t, \overline \xi) \bigr] [\xi - \overline \xi] > 0\;(\xi \neq \overline \xi),\quad a(x,t, \xi) \xi \geq c_ 3 | \xi |^ p$ $$\forall (t, \xi, \overline \xi) \in \mathbb{R}^{2N + 1}$$, a.e. $$x \in \Omega$$, $$f$$ and $$\Phi$$ belong respectively to $$W^{-1,p/(p-1)} (\Omega)$$ and $$(C^ 0(\mathbb{R}))^ N$$ and $$g$$ satisfies some more technical conditions.
It is proven that there exists a renormalized solution to $$(*)$$, i.e. a function $$u$$ of $$W_ 0^{1,p} (\Omega)$$ such that $- \biggl[\text{div} \bigl( a(x,u,Du) \bigr) \biggr] h(u) - \text{div} \bigl( \Phi (u)h(u)\bigr) + \Phi (u)h'(u)Du+ g(x,u)h(u) = fh(u)$ in $${\mathcal D}' (\Omega)$$, $$\forall h \in C^ 1_ c (\Omega)$$. Properties of these solutions are then studied: a sufficient condition for a renormalized solution to be a usual weak solution of $$(*)$$; energy identities; the largest class of possible test functions. $$L^ s$$-regularity of renormalized solutions is at least studied in the general case, and in a case where a further coerciveness condition (a growth condition) on $$g$$ provides more regularity. Application of these two regularity results to the existence of weak solutions is finally given.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000)
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