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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]

P. Amster, M. C. Mariani and O. Mendez, Nonlinear boundary conditions for elliptic equations, Electronic Journal of Differential Equations, 144 (2005), 1-8.  Google Scholar

[2]

J. F. Bonder and J. D. Rossi, Existence Results for the p-Laplacian with nonlinear boundary conditions, J. M. A. A., 263 (2001), 195-223.  Google Scholar

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J. F. Bonder, J. P. Pinasco and J. D. Rossi, Existence results for a Hamiltonian elliptic systems with nonlinear boundary conditions, Electronic Journal of Differential Equations, 40 (1999), 1-15. Google Scholar

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J. F. Bonder and J. D. Rossi, Existence for an elliptic system with nonlinear boundary conditions via fixed point methods, Adv.Differential Equations, 6 (2001), 1-20.  Google Scholar

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M. Chipot, I. Shafrir and M. Fila, On the solutions to some elliptic equations with nonlinear boundary conditions, Adv. Differential Equations, 1 (1996), 91-110.  Google Scholar

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M. Chipot, M. Chlebik, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $R$N+with a nonlinear boundary condition, J. Math. Anal. Appl., 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958.  Google Scholar

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P. Drabek, Nonlinear eigenvalue problems for the p.Laplacian in $R$N, Math. Nachr., 173 (1995), 131-139. doi: 10.1002/mana.19951730109.  Google Scholar

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J. I. Diaz, Nonlinear partial differential equations and free boundaries, Elliptic equations. Pitman Adv. Publ., Boston MA, 323 (1985), 44-95.  Google Scholar

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P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Reg. conf. ser. Math, 65 (1986), 1-100.  Google Scholar

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C. Flores and M. del Pino, Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains, Comm. Partial Differential Equations, 26 (2001), 2189-2210. doi: 10.1081/PDE-100107818.  Google Scholar

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B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations, 7 (1994), 301-313.  Google Scholar

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Y. ILYasov and T. Runst, Positive solutions for indefinite inhomogeneous Neumann elliptic problems, Electronic Journal of Differential Equations, 57 (2003), 1-21.  Google Scholar

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D. A. Kandilakis and A. N. Lyberopoulos, Indefinite quasilinear elliptic problems with subcritical and supercritical nonlinearities on unbounded domains, J. Differential Equations, 230 (2006), 337-361. doi: 10.1016/j.jde.2006.03.008.  Google Scholar

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R. Filippucci, P. Pucci and v. Rădulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. Partial Differential Equations, 33 (2008), 706-717. doi: 10.1080/03605300701518208.  Google Scholar

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X. C. Song, W. H. Wang and P. H. Zhao, Positive solutions of elliptic equations with nonlinear boundary conditions, Nonlinear Analysis, 70 (2009), 328-334. doi: 10.1016/j.na.2007.12.003.  Google Scholar

[19]

S. Z. Song and C. L. Tang, Resonance problems for the p-Laplacian with a nonlinear boundary condition, Nonlinear Analysis, 64 (2006), 2007-2021. doi: 10.1016/j.na.2005.07.035.  Google Scholar

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S. Terraccini, Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Differential Integral Equations, 8 (1995), 1911-1922.  Google Scholar

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J. H. Zhao and P. H. Zhao, Infinitely many weak solutions for a p-Laplacian equation with nonlinear boundary conditions, EJDE, 90 (2007), 1-14.  Google Scholar

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J. H. Zhao and P. H. Zhao, Existence of infinitely many weak solutions for the p-Laplacian with nonlinear boundary conditions, Nonlinear Analysis, 69 (2008), 1343-1355. doi: 10.1016/j.na.2007.06.036.  Google Scholar

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Z. M. Guo and J. R. L. Webb, Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large, Proc. Roy. Soc. Edinburgh, 124A (1994), 189-198.  Google Scholar

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Z. M. Guo, On the number of positive solutions for quasilinear elliptic eigenvalue problems, Nonlinear Analysis, 27 (1996), 229-247. doi: 10.1016/0362-546X(94)00352-I.  Google Scholar

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D. D. Hai, Positive solutions of quasilinear boundary value problems, J. Math.Anal. Appl., 217 (1998), 672-686. doi: 10.1006/jmaa.1997.5762.  Google Scholar

[26]

B. J. Xuan and Z. C. Chen, Solvability of singular quasilinear elliptic equation, Chinese Ann of Math., 20A (1999), 117-128.  Google Scholar

[27]

Z. D. Yang and Q. S. Lu, Existence and multipicity of positive entire solutions for a class of quasilinear elliptic equation, Journal Beijing University of Aeronautics and Astronautics, 27 (2001), 217-220. Google Scholar

[28]

Z. D. Yang, Existence of positive bounded entire solutions for quasilinear elliptic equations, Appl. Math. Comput., 156 (2004), 743-754. doi: 10.1016/j.amc.2003.06.024.  Google Scholar

[29]

M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations, J. Diff. Equs., 76 (1988), 159-189. doi: 10.1016/0022-0396(88)90068-X.  Google Scholar

[30]

Z. D. Yang, Existence of positive entire solutions for singular and non-singular quasi-linear elliptic equation, J. Comm. Appl. Math., 197 (2006), 355-364. doi: 10.1016/j.cam.2005.08.027.  Google Scholar

[31]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar

[32]

S. Kichenassamy and J. Smoller, On the existence of radial solutions of quasilinear elliptic equations, Nonlinearity, 3 (1990), 677-694. doi: 10.1088/0951-7715/3/3/008.  Google Scholar

[33]

A. Mohammed, Positive solutions of the $p$-Laplace equation with singular nonlinearity, J. Math. Anal. Appl., 352 (2009), 234-245. doi: 10.1016/j.jmaa.2008.06.018.  Google Scholar

show all references

References:
[1]

P. Amster, M. C. Mariani and O. Mendez, Nonlinear boundary conditions for elliptic equations, Electronic Journal of Differential Equations, 144 (2005), 1-8.  Google Scholar

[2]

J. F. Bonder and J. D. Rossi, Existence Results for the p-Laplacian with nonlinear boundary conditions, J. M. A. A., 263 (2001), 195-223.  Google Scholar

[3]

J. F. Bonder, J. P. Pinasco and J. D. Rossi, Existence results for a Hamiltonian elliptic systems with nonlinear boundary conditions, Electronic Journal of Differential Equations, 40 (1999), 1-15. Google Scholar

[4]

J. F. Bonder and J. D. Rossi, Existence for an elliptic system with nonlinear boundary conditions via fixed point methods, Adv.Differential Equations, 6 (2001), 1-20.  Google Scholar

[5]

M. Chipot, I. Shafrir and M. Fila, On the solutions to some elliptic equations with nonlinear boundary conditions, Adv. Differential Equations, 1 (1996), 91-110.  Google Scholar

[6]

M. Chipot, M. Chlebik, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $R$N+with a nonlinear boundary condition, J. Math. Anal. Appl., 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958.  Google Scholar

[7]

P. Drabek, Nonlinear eigenvalue problems for the p.Laplacian in $R$N, Math. Nachr., 173 (1995), 131-139. doi: 10.1002/mana.19951730109.  Google Scholar

[8]

J. I. Diaz, Nonlinear partial differential equations and free boundaries, Elliptic equations. Pitman Adv. Publ., Boston MA, 323 (1985), 44-95.  Google Scholar

[9]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Reg. conf. ser. Math, 65 (1986), 1-100.  Google Scholar

[10]

C. Flores and M. del Pino, Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains, Comm. Partial Differential Equations, 26 (2001), 2189-2210. doi: 10.1081/PDE-100107818.  Google Scholar

[11]

Z. M. Guo, Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems, Nonlinear Anal., 18 (1992), 957-971. doi: 10.1016/0362-546X(92)90132-X.  Google Scholar

[12]

B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations, 7 (1994), 301-313.  Google Scholar

[13]

Y. ILYasov and T. Runst, Positive solutions for indefinite inhomogeneous Neumann elliptic problems, Electronic Journal of Differential Equations, 57 (2003), 1-21.  Google Scholar

[14]

D. A. Kandilakis and A. N. Lyberopoulos, Indefinite quasilinear elliptic problems with subcritical and supercritical nonlinearities on unbounded domains, J. Differential Equations, 230 (2006), 337-361. doi: 10.1016/j.jde.2006.03.008.  Google Scholar

[15]

E. Montefusco and V. Radulescu, Nonlinear eigenvalue problems for quasilinear operators on unbounded domains, Nonlinear Differ. Equ. Appl., 8 (2001), 481-497. doi: 10.1007/PL00001460.  Google Scholar

[16]

K. Pflüger, Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary condition, Electronic Journal of Differential Equations, 10 (1998), 1-13.  Google Scholar

[17]

R. Filippucci, P. Pucci and v. Rădulescu, Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. Partial Differential Equations, 33 (2008), 706-717. doi: 10.1080/03605300701518208.  Google Scholar

[18]

X. C. Song, W. H. Wang and P. H. Zhao, Positive solutions of elliptic equations with nonlinear boundary conditions, Nonlinear Analysis, 70 (2009), 328-334. doi: 10.1016/j.na.2007.12.003.  Google Scholar

[19]

S. Z. Song and C. L. Tang, Resonance problems for the p-Laplacian with a nonlinear boundary condition, Nonlinear Analysis, 64 (2006), 2007-2021. doi: 10.1016/j.na.2005.07.035.  Google Scholar

[20]

S. Terraccini, Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Differential Integral Equations, 8 (1995), 1911-1922.  Google Scholar

[21]

J. H. Zhao and P. H. Zhao, Infinitely many weak solutions for a p-Laplacian equation with nonlinear boundary conditions, EJDE, 90 (2007), 1-14.  Google Scholar

[22]

J. H. Zhao and P. H. Zhao, Existence of infinitely many weak solutions for the p-Laplacian with nonlinear boundary conditions, Nonlinear Analysis, 69 (2008), 1343-1355. doi: 10.1016/j.na.2007.06.036.  Google Scholar

[23]

Z. M. Guo and J. R. L. Webb, Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large, Proc. Roy. Soc. Edinburgh, 124A (1994), 189-198.  Google Scholar

[24]

Z. M. Guo, On the number of positive solutions for quasilinear elliptic eigenvalue problems, Nonlinear Analysis, 27 (1996), 229-247. doi: 10.1016/0362-546X(94)00352-I.  Google Scholar

[25]

D. D. Hai, Positive solutions of quasilinear boundary value problems, J. Math.Anal. Appl., 217 (1998), 672-686. doi: 10.1006/jmaa.1997.5762.  Google Scholar

[26]

B. J. Xuan and Z. C. Chen, Solvability of singular quasilinear elliptic equation, Chinese Ann of Math., 20A (1999), 117-128.  Google Scholar

[27]

Z. D. Yang and Q. S. Lu, Existence and multipicity of positive entire solutions for a class of quasilinear elliptic equation, Journal Beijing University of Aeronautics and Astronautics, 27 (2001), 217-220. Google Scholar

[28]

Z. D. Yang, Existence of positive bounded entire solutions for quasilinear elliptic equations, Appl. Math. Comput., 156 (2004), 743-754. doi: 10.1016/j.amc.2003.06.024.  Google Scholar

[29]

M. Guedda and L. Veron, Local and global properties of solutions of quasilinear elliptic equations, J. Diff. Equs., 76 (1988), 159-189. doi: 10.1016/0022-0396(88)90068-X.  Google Scholar

[30]

Z. D. Yang, Existence of positive entire solutions for singular and non-singular quasi-linear elliptic equation, J. Comm. Appl. Math., 197 (2006), 355-364. doi: 10.1016/j.cam.2005.08.027.  Google Scholar

[31]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar

[32]

S. Kichenassamy and J. Smoller, On the existence of radial solutions of quasilinear elliptic equations, Nonlinearity, 3 (1990), 677-694. doi: 10.1088/0951-7715/3/3/008.  Google Scholar

[33]

A. Mohammed, Positive solutions of the $p$-Laplace equation with singular nonlinearity, J. Math. Anal. Appl., 352 (2009), 234-245. doi: 10.1016/j.jmaa.2008.06.018.  Google Scholar

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