May  2015, 35(5): 2067-2078. doi: 10.3934/dcds.2015.35.2067

Wolff type potential estimates and application to nonlinear equations with negative exponents

1. 

Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023

Received  December 2013 Revised  September 2014 Published  December 2014

In this paper, we are concerned with the positive continuous entire solutions of the Wolff type integral equation $$ u(x)=c(x)W_{\beta,\gamma}(u^{-p})(x), \quad u>0 ~in~ R^n, $$ where $n \geq 1$, $p>0$, $\gamma>1$, $\beta>0$ and $\beta\gamma \neq n$. In addition, $c(x)$ is a double bounded function. Such an integral equation is related to the study of the conformal geometry and nonlinear PDEs, such as $\gamma$-Laplace equations and $k$-Hessian equations with negative exponents. By some Wolff type potential integral estimates, we obtain the asymptotic rates and the integrability of positive solutions, and discuss the existence and nonexistence results of the radial solutions.
Citation: Yutian Lei. Wolff type potential estimates and application to nonlinear equations with negative exponents. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2067-2078. doi: 10.3934/dcds.2015.35.2067
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show all references

References:
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Potential Anal., 16 (2002), 347-372. doi: 10.1023/A:1014845728367.  Google Scholar

[2]

Discrete Contin. Dyn. Syst., 34 (2014), 1879-1904. doi: 10.3934/dcds.2014.34.1879.  Google Scholar

[3]

Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[4]

Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083.  Google Scholar

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Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

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J. Differential Equations, 246 (2009), 216-234. doi: 10.1016/j.jde.2008.06.027.  Google Scholar

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Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar

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Discrete Contin. Dyn. Syst., 34 (2014), 2561-2580. doi: 10.3934/dcds.2014.34.2561.  Google Scholar

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Ann. Inst. Fourier (Grenobel), 33 (1983), 161-187. doi: 10.5802/aif.944.  Google Scholar

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Ann. Seuola Norm. Sup. Pisa, Cl. Sci., 19 (1992), 591-613.  Google Scholar

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

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Potential Anal., 35 (2011), 387-402. doi: 10.1007/s11118-010-9218-5.  Google Scholar

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Discrete Contin. Dyn. Syst., 35 (2015), 1039-1057. doi: 10.3934/dcds.2015.35.1039.  Google Scholar

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J. Differential Equations, 252 (2012), 2739-2758. doi: 10.1016/j.jde.2011.10.009.  Google Scholar

[18]

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[19]

J. Eur. Math. Soc., 6 (2004), 153-180. doi: 10.4171/JEMS/6.  Google Scholar

[20]

Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[21]

Forum. Math., 22 (2010), 1061-1087. doi: 10.1515/forum.2010.057.  Google Scholar

[22]

Adv. Math., 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[23]

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Calc. Var. Partial Differential Equations, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.  Google Scholar

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