September  2011, 16(2): 669-685. doi: 10.3934/dcdsb.2011.16.669

Optimal regularity for $A$-harmonic type equations under the natural growth

1. 

Department of Mathematics, Beijing Jiaotong University, Beijing 100044

2. 

Department of Mathematics, Taizhou University, Linhai, Zhejiang 317000, China

3. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539

Received  January 2010 Revised  October 2010 Published  June 2011

In this paper we are concerned with a class of nonlinear degenerate elliptic equations under the natural growth. We show that each bounded weak solution of $A$-harmonic type equations under the natural growth belongs to local Hölder continuity based on a density lemma and the Moser-Nash's argument. Then we show that its weak solution is of optimal regularity with the Hölder exponent for any $\gamma$: $0\le \gamma<\kappa$, where $\kappa$ is the same as the Hölder's index for homogeneous $A$-harmonic equations.
Citation: Shenzhou Zheng, Xueliang Zheng, Zhaosheng Feng. Optimal regularity for $A$-harmonic type equations under the natural growth. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 669-685. doi: 10.3934/dcdsb.2011.16.669
References:
[1]

J. Math. Anal. Appl., 140 (1989), 115-135.  Google Scholar

[2]

Differential and Integral Equations, 12 (1999), 261-274.  Google Scholar

[3]

Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[4]

Nonlinear Anal., 51 (2002), 783-800. doi: 10.1016/S0362-546X(01)00862-8.  Google Scholar

[5]

Indiana Univ. Math. J., 53 (2004), 683-718. doi: 10.1512/iumj.2004.53.2462.  Google Scholar

[6]

Differential and Integral Equations, 21 (2008), 717-741.  Google Scholar

[7]

Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 397-413. Google Scholar

[8]

Discrete Contin. Dyn. Syst., 24 (2009), 763-780. doi: 10.3934/dcds.2009.24.763.  Google Scholar

[9]

Ann. Mat. Pura. Appl., 155 (1989), 1-24. doi: 10.1007/BF01765932.  Google Scholar

[10]

Spinger-Verlag, Berlin, 2001.  Google Scholar

[11]

Annals of Mathematics Studies, 105, Princeton Univ. Press, 1983.  Google Scholar

[12]

Manuscripta Math., 57 (1986), 55-99. doi: 10.1007/BF01172492.  Google Scholar

[13]

Oxford University Press, New York, 1993.  Google Scholar

[14]

Ark. Mat., 26 (1988), 87-105. doi: 10.1007/BF02386110.  Google Scholar

[15]

American Mathematical Society, Providence, Rhode Island, 1997.  Google Scholar

[16]

Pitman Res. Notes Math. Ser. Harlow, 314 (1994), 58-64.  Google Scholar

[17]

J. Reine Angew Math., 454 (1994), 143-161. doi: 10.1515/crll.1994.454.143.  Google Scholar

[18]

Proceedings of the International Congress of Mathematicians, 167-176, Vol. III, Higher Ed. Press, Beijing, 2002.  Google Scholar

[19]

Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar

[20]

Ark. Mat., 31 (1993), 339-353. doi: 10.1007/BF02559490.  Google Scholar

[21]

Acta Math., 155 (1985), 153-171. doi: 10.1007/BF02392541.  Google Scholar

[22]

Science Press House, Beijing, 1985. Google Scholar

[23]

Illinois J. Math., 43 (1999), 613-632.  Google Scholar

[24]

Amer. Math. Soc. (Translation of Math Monographs), Vol. 73, Providence, 1989.  Google Scholar

[25]

Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.  Google Scholar

[26]

Amer. J. Math., 124 (2002), 369-410. doi: 10.1353/ajm.2002.0012.  Google Scholar

[27]

Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.  Google Scholar

[28]

Acta Math. Sinica, 20 (2004), 193-205. doi: 10.1007/s10114-003-0250-x.  Google Scholar

[29]

Acta Math. Appl. Sinica, 20 (2004), 115-122. doi: 10.1007/s10255-004-0154-2.  Google Scholar

[30]

J. Math. Anal. Appl., 346 (2008), 359-373. doi: 10.1016/j.jmaa.2008.05.059.  Google Scholar

show all references

References:
[1]

J. Math. Anal. Appl., 140 (1989), 115-135.  Google Scholar

[2]

Differential and Integral Equations, 12 (1999), 261-274.  Google Scholar

[3]

Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[4]

Nonlinear Anal., 51 (2002), 783-800. doi: 10.1016/S0362-546X(01)00862-8.  Google Scholar

[5]

Indiana Univ. Math. J., 53 (2004), 683-718. doi: 10.1512/iumj.2004.53.2462.  Google Scholar

[6]

Differential and Integral Equations, 21 (2008), 717-741.  Google Scholar

[7]

Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 397-413. Google Scholar

[8]

Discrete Contin. Dyn. Syst., 24 (2009), 763-780. doi: 10.3934/dcds.2009.24.763.  Google Scholar

[9]

Ann. Mat. Pura. Appl., 155 (1989), 1-24. doi: 10.1007/BF01765932.  Google Scholar

[10]

Spinger-Verlag, Berlin, 2001.  Google Scholar

[11]

Annals of Mathematics Studies, 105, Princeton Univ. Press, 1983.  Google Scholar

[12]

Manuscripta Math., 57 (1986), 55-99. doi: 10.1007/BF01172492.  Google Scholar

[13]

Oxford University Press, New York, 1993.  Google Scholar

[14]

Ark. Mat., 26 (1988), 87-105. doi: 10.1007/BF02386110.  Google Scholar

[15]

American Mathematical Society, Providence, Rhode Island, 1997.  Google Scholar

[16]

Pitman Res. Notes Math. Ser. Harlow, 314 (1994), 58-64.  Google Scholar

[17]

J. Reine Angew Math., 454 (1994), 143-161. doi: 10.1515/crll.1994.454.143.  Google Scholar

[18]

Proceedings of the International Congress of Mathematicians, 167-176, Vol. III, Higher Ed. Press, Beijing, 2002.  Google Scholar

[19]

Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar

[20]

Ark. Mat., 31 (1993), 339-353. doi: 10.1007/BF02559490.  Google Scholar

[21]

Acta Math., 155 (1985), 153-171. doi: 10.1007/BF02392541.  Google Scholar

[22]

Science Press House, Beijing, 1985. Google Scholar

[23]

Illinois J. Math., 43 (1999), 613-632.  Google Scholar

[24]

Amer. Math. Soc. (Translation of Math Monographs), Vol. 73, Providence, 1989.  Google Scholar

[25]

Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.  Google Scholar

[26]

Amer. J. Math., 124 (2002), 369-410. doi: 10.1353/ajm.2002.0012.  Google Scholar

[27]

Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.  Google Scholar

[28]

Acta Math. Sinica, 20 (2004), 193-205. doi: 10.1007/s10114-003-0250-x.  Google Scholar

[29]

Acta Math. Appl. Sinica, 20 (2004), 115-122. doi: 10.1007/s10255-004-0154-2.  Google Scholar

[30]

J. Math. Anal. Appl., 346 (2008), 359-373. doi: 10.1016/j.jmaa.2008.05.059.  Google Scholar

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