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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.

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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