## Equivalence between the growth of $$\int_{b(x,r)}| \nabla u| ^ pdy$$ and T in the equation $$P[u]=T$$.(English)Zbl 0707.35033

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 $\int_{B(x,R)}| \nabla u|^ p dy\leq C\cdot R^{\sigma}.$
Reviewer: Y.Suyama

### MSC:

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

### Keywords:

Morrey spaces; monotonicity; coerciveness
Full Text:

### References:

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