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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

 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.
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.   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.   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.   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.   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.   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.  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.  doi: 10.1002/mana.19951730109.  Google Scholar [8] J. I. Diaz, Nonlinear partial differential equations and free boundaries,, Elliptic equations. Pitman Adv. Publ., 323 (1985), 44.   Google Scholar [9] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, Reg. conf. ser. Math, 65 (1986), 1.   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.  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.  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.   Google Scholar [13] Y. ILYasov and T. Runst, Positive solutions for indefinite inhomogeneous Neumann elliptic problems,, Electronic Journal of Differential Equations, 57 (2003), 1.   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.  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.  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.   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.  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.  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.  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.   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.   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.  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.   Google Scholar [24] Z. M. Guo, On the number of positive solutions for quasilinear elliptic eigenvalue problems,, Nonlinear Analysis, 27 (1996), 229.  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.  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.   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.   Google Scholar [28] Z. D. Yang, Existence of positive bounded entire solutions for quasilinear elliptic equations,, Appl. Math. Comput., 156 (2004), 743.  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.  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.  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.  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.  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.  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.   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.   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.   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.   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.   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.  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.  doi: 10.1002/mana.19951730109.  Google Scholar [8] J. I. Diaz, Nonlinear partial differential equations and free boundaries,, Elliptic equations. Pitman Adv. Publ., 323 (1985), 44.   Google Scholar [9] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, Reg. conf. ser. Math, 65 (1986), 1.   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.  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.  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.   Google Scholar [13] Y. ILYasov and T. Runst, Positive solutions for indefinite inhomogeneous Neumann elliptic problems,, Electronic Journal of Differential Equations, 57 (2003), 1.   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.  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.  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.   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.  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.  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.  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.   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.   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.  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.   Google Scholar [24] Z. M. Guo, On the number of positive solutions for quasilinear elliptic eigenvalue problems,, Nonlinear Analysis, 27 (1996), 229.  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.  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.   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.   Google Scholar [28] Z. D. Yang, Existence of positive bounded entire solutions for quasilinear elliptic equations,, Appl. Math. Comput., 156 (2004), 743.  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.  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.  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.  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.  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.  doi: 10.1016/j.jmaa.2008.06.018.  Google Scholar
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