Positive solutions of p-Laplacian equations with nonlinear boundary condition

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[6]	M. Chipot, M. Chlebik, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $\mathbb{R}^{N}$^N₊with a nonlinear boundary condition, J. Math. Anal. Appl., 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958.
[7]	P. Drabek, Nonlinear eigenvalue problems for the p.Laplacian in $\mathbb{R}^{N}$^N, Math. Nachr., 173 (1995), 131-139. doi: 10.1002/mana.19951730109.
[8]	J. I. Diaz, Nonlinear partial differential equations and free boundaries, Elliptic equations. Pitman Adv. Publ., Boston MA, 323 (1985), 44-95.
[9]	P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Reg. conf. ser. Math, 65 (1986), 1-100.
[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.
[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.
[12]	B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations, 7 (1994), 301-313.
[13]	Y. ILYasov and T. Runst, Positive solutions for indefinite inhomogeneous Neumann elliptic problems, Electronic Journal of Differential Equations, 57 (2003), 1-21.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	S. Terraccini, Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Differential Integral Equations, 8 (1995), 1911-1922.
[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.
[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.
[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.
[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.
[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.
[26]	B. J. Xuan and Z. C. Chen, Solvability of singular quasilinear elliptic equation, Chinese Ann of Math., 20A (1999), 117-128.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[6]	M. Chipot, M. Chlebik, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $\mathbb{R}^{N}$^N₊with a nonlinear boundary condition, J. Math. Anal. Appl., 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958.
[7]	P. Drabek, Nonlinear eigenvalue problems for the p.Laplacian in $\mathbb{R}^{N}$^N, Math. Nachr., 173 (1995), 131-139. doi: 10.1002/mana.19951730109.
[8]	J. I. Diaz, Nonlinear partial differential equations and free boundaries, Elliptic equations. Pitman Adv. Publ., Boston MA, 323 (1985), 44-95.
[9]	P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Reg. conf. ser. Math, 65 (1986), 1-100.
[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.
[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.
[12]	B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations, 7 (1994), 301-313.
[13]	Y. ILYasov and T. Runst, Positive solutions for indefinite inhomogeneous Neumann elliptic problems, Electronic Journal of Differential Equations, 57 (2003), 1-21.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	S. Terraccini, Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Differential Integral Equations, 8 (1995), 1911-1922.
[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.
[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.
[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.
[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.
[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.
[26]	B. J. Xuan and Z. C. Chen, Solvability of singular quasilinear elliptic equation, Chinese Ann of Math., 20A (1999), 117-128.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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