## 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:

 [1] Adams, D., Traces of potentials arising from translation invariant operators, Ann. Scuola Norm. Sup. Pisa, 25, 203-217 (1971) · Zbl 0219.46027 [2] Adams, R., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0314.46030 [3] Campanato, S., Equazioni ellitiche del II∘ ordine e spazi $$L^{2, 1}$$, Ann. Mat. Pura Appl. (4), 69, 321-381 (1965) · Zbl 0145.36603 [4] DeGiorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (3), 3, 25-43 (1957) · Zbl 0084.31901 [5] Dibenedetto, E.; Trudinger, N., Harnack inequalities for quasi-minima of variational integrals (1984), I.H.P · Zbl 0565.35012 [6] Gariepy, R.; Ziemer, W. P., Behaviour at the boundary of solutions of quasilinear elliptic equations, Arch. Rational Mech. Anal., 56, 372-384 (1972) · Zbl 0297.35032 [7] Giaquinta, M., Multiple Integrals in the Calculus of Variations and Nonlinear Systems (1983), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0516.49003 [8] Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer New York · Zbl 0562.35001 [9] Hartman, P.; Stampacchia, G., On some non-linear elliptic differential functional equations, Acta Math., 115, 271-310 (1966) · Zbl 0142.38102 [10] Hedberg, L.; Wolff, T., Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble), 23, 161-187 (1983) · Zbl 0508.31008 [11] Kadlec; Necas, J., Sulla regolarita delle soluzioni di equazioni ellittiche negli spazi $$H^{k,l}$$, Ann. Scuola Norm. Sup. Pisa (3), 21, 527 (1967) · Zbl 0157.42203 [12] Ladyzenskaya, O. A.; Ural’tseva, N. N., Linear and Quasilinear Elliptic Equation (1968), Academic Press: Academic Press New York · Zbl 0164.13002 [13] Leray, J.; Lions, J. L., Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93, 97-107 (1965) · Zbl 0132.10502 [14] Lewy, H.; Stampacchia, G., On the smoothness of superharmonics which solve the minimum problem, J. Analyse Math., 23, 227-236 (1970) · Zbl 0206.40702 [15] Lions, J. L., Quelques methodes de résolution des problèmes aux limites non linéaires (1969), Dunod: Dunod Paris · Zbl 0189.40603 [16] Maz’ja, V. G., Sobolev Spaces (1986), Springer-Verlag: Springer-Verlag Berlin [17] Morrey, C. B., Multiple Integrals in the Calculus of Variations (1966), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0142.38701 [18] Moser, J., A new proof of DeGiorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13, 457-468 (1960) · Zbl 0111.09301 [19] Rakotoson, J. M., Réarrangement relatif dans les équations elliptiques quasilineaires avec un second membre distribution: Application à un théorème d’existence et de régularité, J. Differential Equations, 66, 391-419 (1987) · Zbl 0652.35041 [21] Rakotoson, J. M.; Temam, R., Relative rearrangement in quasilinear variational elliptic inequalities, Indiana Univ. Math. J., 36 (1987) [22] Rodrigues, J.-F, Obstacle problems in mathematical physics, (Mathematics Studies, Vol. 134 (1987), North-Holland: North-Holland Amsterdam) · Zbl 0606.73017 [23] Rakotoson, J. M.; Ziemer, W. P., Local behavior of solutions of quasilinear elliptic equations with general structure, (preprint of The Institute for Applied Mathematics and Scientific Computing, No. 8810 (Indiana University). preprint of The Institute for Applied Mathematics and Scientific Computing, No. 8810 (Indiana University), T.A.M.S., Vol. 317 (Jan. 1990)), No. 1 · Zbl 0708.35023 [24] Sakai, M., Solutions to the obstacle problem as green potentials, J. Anal. Math., 44, 97-116 (1984) · Zbl 0577.49005 [25] Sobolev, S. L., Applications of Functional Analysis in Mathematical Physics (1963), Amer. Math. Soc: Amer. Math. Soc Providence, RI · Zbl 0123.09003 [26] Serrin, J., Local behavior of solutions of quasilinear equations, Acta Math., 111, 247-302 (1964) · Zbl 0128.09101 [27] Trudinger, N., On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math., 20, 721-747 (1967) · Zbl 0153.42703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.