Equivalence between the growth of \(\int_{b(x,r)}| \nabla u| ^ pdy\) and T in the equation \(P[u]=T\)
Let us define \[ (I)\;M_{\lambda}^{-1,q}(\Omega)=\{T\in W^{- 1,q}(\Omega)\text{ such that }\sup_{k}[\rho^{-(\lambda /q)}\| T\|_{\bar W^{-1,q}}(\Omega (x,\rho))]<\infty \}, \] where \(\Omega (x,\rho)=\Omega \cap B(x,\rho)\), \(K=\{(\rho,x)\in I\times \Omega \}\), \(I=(0\), diameter of \(\Omega\)), \[ (II)\;Au+F(u,\nabla u)=T, \] where (i) \(\Omega\) is an open set in \({\mathbb{R}}^ N\), (ii) for each \(u\in W^{1,p}_{loc}(\Omega)\cap L^{\infty}_{loc}(\Omega)\), \(Au=- \sum^{N}_{i=1}(\partial /\partial x_ i)a_ i\) (x,u,\(\nabla u)\), \(a_ i\) are Borelian functions from \(\Omega \times {\mathbb{R}}\times {\mathbb{R}}^ N\) into R, and for a.e. \(x\in \Omega\) and all \((\eta,\xi)\in {\mathbb{R}}\times {\mathbb{R}}^ N\), \[ | a_ i(x,\eta,\xi)| \leq a(| \eta |)[| \xi |^{p-1}+a_ 0(x)],\;a_ 0\in L_{loc}^{q,N-p+\beta}(\Omega),\;\beta >0 \] and a is an increasing function \(R_+\to R_+\) and \(a_ i\) satisfies some monotonicity and coerciveness, and (iii) the nonlinearity F is such that for a.e. \(x\in \Omega\), \(\forall (\eta,\xi)\in {\mathbb{R}}\times {\mathbb{R}}^ N\), \[ | F(x,\eta,\xi)| \leq f(| \eta |)[| \xi |^{p- \gamma}+f_ 0(x)], \] where \(\gamma >0\), \(f_ 0\in L_{loc}^{1,N- p+\sigma}(\Omega)\), \(\sigma >0\) and f is an increasing function \({\mathbb{R}}_+\to {\mathbb{R}}_+.\) If u is a local solution of \(Au+F(u,\nabla u)=T\), then \(T\in M^{- 1,q}_{\lambda,loc}(\Omega)\) for \(\lambda >N-p\), \(1/p+1/q=1\) if and only if for all subsets \(\Omega'\) relatively compact, there exist \(C>0\) and \(\sigma>N-p\) such that \(\forall x\in \Omega'\), \(\forall R>0\); \(B(x,2R)\subset \Omega'_ 0\) we have \[ \int_{B(x,R)}| \nabla u|^ p dy\leq C\cdot R^{\sigma}. \]
scientific article; zbMATH DE number 4161199
scientific article