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September  2011, 16(2): 623-636. doi: 10.3934/dcdsb.2011.16.623

Positive solutions of $p$-Laplacian equations with nonlinear boundary condition

Zuodong Yang 1, , Jing Mo 1, and  Subei Li 2,

1. 

Institute of Mathematics, School of Mathematics Science, Nanjing Normal University, Jiangsu Nanjing 210046, China, China
2. 

School of Mathematics and Physics, Xuzhou Institute of Technology, Xuzhou, Jiangsu 221111, China

Received  November 2009 Revised  February 2011 Published  June 2011

In this paper we study the following problem $$-\triangle_{p}u+|u|^{p-2}u=f(x,u) $$ in a bounded smooth domain $\Omega \subset {\bf R}^{N}$ with a nonlinear boundary value condition $|\nabla u|^{p-2}\frac{\partial u}{\partial\nu}=g(x,u)$. Results on the existence of positive solutions are obtained by the sub-supersolution method and the Mountain Pass Lemma.
Keywords: sub-supersolution, Mountain Pass Theorem., positive solution, Nonlinear boundary value condition.
Mathematics Subject Classification: Primary: 35J65, 35J50; Secondary: 35J5.
Citation: Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623
References:
[1]

Electronic Journal of Differential Equations, 144 (2005), 1-8.  Google Scholar

[2]

J. M. A. A., 263 (2001), 195-223.  Google Scholar

[3]

Electronic Journal of Differential Equations, 40 (1999), 1-15. Google Scholar

[4]

Adv.Differential Equations, 6 (2001), 1-20.  Google Scholar

[5]

Adv. Differential Equations, 1 (1996), 91-110.  Google Scholar

[6]

J. Math. Anal. Appl., 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958.  Google Scholar

[7]

Math. Nachr., 173 (1995), 131-139. doi: 10.1002/mana.19951730109.  Google Scholar

[8]

Elliptic equations. Pitman Adv. Publ., Boston MA, 323 (1985), 44-95.  Google Scholar

[9]

Reg. conf. ser. Math, 65 (1986), 1-100.  Google Scholar

[10]

Comm. Partial Differential Equations, 26 (2001), 2189-2210. doi: 10.1081/PDE-100107818.  Google Scholar

[11]

Nonlinear Anal., 18 (1992), 957-971. doi: 10.1016/0362-546X(92)90132-X.  Google Scholar

[12]

Differential Integral Equations, 7 (1994), 301-313.  Google Scholar

[13]

Electronic Journal of Differential Equations, 57 (2003), 1-21.  Google Scholar

[14]

J. Differential Equations, 230 (2006), 337-361. doi: 10.1016/j.jde.2006.03.008.  Google Scholar

[15]

Nonlinear Differ. Equ. Appl., 8 (2001), 481-497. doi: 10.1007/PL00001460.  Google Scholar

[16]

Electronic Journal of Differential Equations, 10 (1998), 1-13.  Google Scholar

[17]

Comm. Partial Differential Equations, 33 (2008), 706-717. doi: 10.1080/03605300701518208.  Google Scholar

[18]

Nonlinear Analysis, 70 (2009), 328-334. doi: 10.1016/j.na.2007.12.003.  Google Scholar

[19]

Nonlinear Analysis, 64 (2006), 2007-2021. doi: 10.1016/j.na.2005.07.035.  Google Scholar

[20]

Differential Integral Equations, 8 (1995), 1911-1922.  Google Scholar

[21]

EJDE, 90 (2007), 1-14.  Google Scholar

[22]

Nonlinear Analysis, 69 (2008), 1343-1355. doi: 10.1016/j.na.2007.06.036.  Google Scholar

[23]

Proc. Roy. Soc. Edinburgh, 124A (1994), 189-198.  Google Scholar

[24]

Nonlinear Analysis, 27 (1996), 229-247. doi: 10.1016/0362-546X(94)00352-I.  Google Scholar

[25]

J. Math.Anal. Appl., 217 (1998), 672-686. doi: 10.1006/jmaa.1997.5762.  Google Scholar

[26]

Chinese Ann of Math., 20A (1999), 117-128.  Google Scholar

[27]

Journal Beijing University of Aeronautics and Astronautics, 27 (2001), 217-220. Google Scholar

[28]

Appl. Math. Comput., 156 (2004), 743-754. doi: 10.1016/j.amc.2003.06.024.  Google Scholar

[29]

J. Diff. Equs., 76 (1988), 159-189. doi: 10.1016/0022-0396(88)90068-X.  Google Scholar

[30]

J. Comm. Appl. Math., 197 (2006), 355-364. doi: 10.1016/j.cam.2005.08.027.  Google Scholar

[31]

Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar

[32]

Nonlinearity, 3 (1990), 677-694. doi: 10.1088/0951-7715/3/3/008.  Google Scholar

[33]

J. Math. Anal. Appl., 352 (2009), 234-245. doi: 10.1016/j.jmaa.2008.06.018.  Google Scholar

show all references

References:
[1]

Electronic Journal of Differential Equations, 144 (2005), 1-8.  Google Scholar

[2]

J. M. A. A., 263 (2001), 195-223.  Google Scholar

[3]

Electronic Journal of Differential Equations, 40 (1999), 1-15. Google Scholar

[4]

Adv.Differential Equations, 6 (2001), 1-20.  Google Scholar

[5]

Adv. Differential Equations, 1 (1996), 91-110.  Google Scholar

[6]

J. Math. Anal. Appl., 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958.  Google Scholar

[7]

Math. Nachr., 173 (1995), 131-139. doi: 10.1002/mana.19951730109.  Google Scholar

[8]

Elliptic equations. Pitman Adv. Publ., Boston MA, 323 (1985), 44-95.  Google Scholar

[9]

Reg. conf. ser. Math, 65 (1986), 1-100.  Google Scholar

[10]

Comm. Partial Differential Equations, 26 (2001), 2189-2210. doi: 10.1081/PDE-100107818.  Google Scholar

[11]

Nonlinear Anal., 18 (1992), 957-971. doi: 10.1016/0362-546X(92)90132-X.  Google Scholar

[12]

Differential Integral Equations, 7 (1994), 301-313.  Google Scholar

[13]

Electronic Journal of Differential Equations, 57 (2003), 1-21.  Google Scholar

[14]

J. Differential Equations, 230 (2006), 337-361. doi: 10.1016/j.jde.2006.03.008.  Google Scholar

[15]

Nonlinear Differ. Equ. Appl., 8 (2001), 481-497. doi: 10.1007/PL00001460.  Google Scholar

[16]

Electronic Journal of Differential Equations, 10 (1998), 1-13.  Google Scholar

[17]

Comm. Partial Differential Equations, 33 (2008), 706-717. doi: 10.1080/03605300701518208.  Google Scholar

[18]

Nonlinear Analysis, 70 (2009), 328-334. doi: 10.1016/j.na.2007.12.003.  Google Scholar

[19]

Nonlinear Analysis, 64 (2006), 2007-2021. doi: 10.1016/j.na.2005.07.035.  Google Scholar

[20]

Differential Integral Equations, 8 (1995), 1911-1922.  Google Scholar

[21]

EJDE, 90 (2007), 1-14.  Google Scholar

[22]

Nonlinear Analysis, 69 (2008), 1343-1355. doi: 10.1016/j.na.2007.06.036.  Google Scholar

[23]

Proc. Roy. Soc. Edinburgh, 124A (1994), 189-198.  Google Scholar

[24]

Nonlinear Analysis, 27 (1996), 229-247. doi: 10.1016/0362-546X(94)00352-I.  Google Scholar

[25]

J. Math.Anal. Appl., 217 (1998), 672-686. doi: 10.1006/jmaa.1997.5762.  Google Scholar

[26]

Chinese Ann of Math., 20A (1999), 117-128.  Google Scholar

[27]

Journal Beijing University of Aeronautics and Astronautics, 27 (2001), 217-220. Google Scholar

[28]

Appl. Math. Comput., 156 (2004), 743-754. doi: 10.1016/j.amc.2003.06.024.  Google Scholar

[29]

J. Diff. Equs., 76 (1988), 159-189. doi: 10.1016/0022-0396(88)90068-X.  Google Scholar

[30]

J. Comm. Appl. Math., 197 (2006), 355-364. doi: 10.1016/j.cam.2005.08.027.  Google Scholar

[31]

Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar

[32]

Nonlinearity, 3 (1990), 677-694. doi: 10.1088/0951-7715/3/3/008.  Google Scholar

[33]

J. Math. Anal. Appl., 352 (2009), 234-245. doi: 10.1016/j.jmaa.2008.06.018.  Google Scholar

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