American Institute of Mathematical Sciences

March  2013, 12(2): 785-802. doi: 10.3934/cpaa.2013.12.785

Multiplicity results for a class of elliptic problems with nonlinear boundary condition

 1 Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin

Received  August 2011 Revised  December 2011 Published  September 2012

This paper provides multiplicity results for a class of nonlinear elliptic problems under a nonhomogeneous Neumann boundary condition. We prove the existence of three nontrivial solutions to these problems which depend on the Fučík spectrum of the negative $p$-Laplacian with a Robin boundary condition. Using variational and topological arguments combined with an equivalent norm on the Sobolev space $W^{1,p}$ it is obtained a smallest positive solution, a greatest negative solution, and a sign-changing solution.
Citation: Patrick Winkert. Multiplicity results for a class of elliptic problems with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (2) : 785-802. doi: 10.3934/cpaa.2013.12.785
References:
 [1] E. A. M. Abreu, J. Marcos do Ó and E. S. Medeiros, Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems,, Nonlinear Anal., 60 (2005), 1443.  doi: 10.1016/j.na.2004.09.058.  Google Scholar [2] R. P. Agarwal and L. Wei, Existence of solutions to nonlinear Neumann boundary value problems with generalized $p$-Laplacian operator, , Comput. Math. Appl., 56 (2008), 530.  doi: 10.1016/j.camwa.2008.01.013.  Google Scholar [3] M. Arias, J. Campos and J.-P. Gossez, On the antimaximum principle and the Fučik spectrum for the Neumann $p$-Laplacian, , Differential Integral Equations, 13 (2000), 217.   Google Scholar [4] D. Averna and G. Bonanno, Three solutions for a Neumann boundary value problem involving the $p$-Laplacian, , Matematiche (Catania), 60 (2005), 81.   Google Scholar [5] G. Bonanno and P. Candito, Three solutions to a Neumann problem for elliptic equations involving the $p$-Laplacian, , Arch. Math. (Basel), 80 (2003), 424.  doi: 10.1007/s00013-003-0479-8.  Google Scholar [6] G. Bonanno and G. D'Aguì, On the Neumann problem for elliptic equations involving the $p$-Laplacian, , J. Math. Anal. Appl., 358 (2009), 223.  doi: 10.1016/j.jmaa.2009.04.055.  Google Scholar [7] S. Carl, Existence and comparison results for noncoercive and nonmonotone multivalued elliptic problems,, Nonlinear Anal., 65 (2006), 1532.  doi: 10.1016/j.na.2005.10.028.  Google Scholar [8] S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities,'', Springer, (2007).   Google Scholar [9] S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems,, Nonlinear Anal., 68 (2008), 2668.  doi: 10.1016/j.na.2007.02.013.  Google Scholar [10] S. Carl, and D. Motreanu, Sign-changing and extremal constant-sign solutions of nonlinear elliptic problems with supercritical nonlinearities,, Comm. Appl. Nonlinear Anal., 14 (2007), 85.   Google Scholar [11] S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian, , Abstr. Appl. Anal., 7 (2002), 613.  doi: 10.1155/S1085337502207010.  Google Scholar [12] S.-G. Deng, Positive solutions for Robin problem involving the $p(x)$-Laplacian, , J. Math. Anal. Appl., 360 (2009), 548.  doi: 10.1016/j.jmaa.2009.06.032.  Google Scholar [13] J. Fernández Bonder and J. D. Rossi, Existence results for the $p$-Laplacian with nonlinear boundary conditions, , J. Math. Anal. Appl., (2001), 195.  doi: 10.1006/jmaa.2001.7609.  Google Scholar [14] J. Fernández Bonder, Multiple positive solutions for quasilinear elliptic problems with sign-changing nonlinearities,, Abstr. Appl. Anal., 12 (2004), 1047.  doi: 10.1155/S1085337504403078.  Google Scholar [15] J. Fernández Bonder, Multiple solutions for the $p$-Laplace equation with nonlinear boundary conditions, , Electron. J. Differential Equations, 37 (2006).   Google Scholar [16] L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,'', Series in Mathematical Analysis and Applications, (2005).   Google Scholar [17] Z. Jin, Multiple solutions for a class of semilinear elliptic equations,, Proc. Amer. Math. Soc., 125 (1997), 3659.  doi: 10.1090/S0002-9939-97-04199-3.  Google Scholar [18] A. Lê, Eigenvalue problems for the $p$-Laplacian, , Nonlinear Anal., 64 (2006), 1057.  doi: 10.1016/j.na.2005.05.056.  Google Scholar [19] C. Li and S. Li, Multiple solutions and sign-changing solutions of a class of nonlinear elliptic equations with Neumann boundary condition,, J. Math. Anal. Appl., 298 (2004), 14.  doi: 10.1016/j.jmaa.2004.01.017.  Google Scholar [20] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar [21] C. Liu and Y. Zheng, Linking solutions for $p$-Laplace equations with nonlinear boundary conditions and indefinite weight, , Calc. Var. Partial Differential Equations, 41 (2011), 261.  doi: 10.1007/s00526-010-0361-z.  Google Scholar [22] S. R. Martínez and J. D. and Rossi, On the Fučik spectrum and a resonance problem for the {$p$-Laplacian with a nonlinear boundary condition, , Nonlinear Anal., 59 (2004), 813.  doi: 10.1016/j.na.2004.07.039.  Google Scholar [23] S. R. Martínez and J. D. Rossi, Weak solutions for the $p$-Laplacian with a nonlinear boundary condition at resonance, , Electron. J. Differential Equations, 27 (2003).   Google Scholar [24] D. Motreanu and P. Winkert, On the Fučik Spectrum for the $p$-Laplacian with a Robin boundary condition,, Nonlinear Anal., 74 (2011), 4671.  doi: 10.1016/j.na.2011.04.033.  Google Scholar [25] D. Motreanu and P. Winkert, The Fučik spectrum for the negative $p$-Laplacian with different boundary conditions,, Springer Optimization and Its Applications, (2012), 471.  doi: 10.1007/978-1-4614-3498-6_28.  Google Scholar [26] P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Regional Conference Series in Mathematics, (1986).   Google Scholar [27] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar [28] P. Winkert, "Comparison Principles and Multiple Solutions for Nonlinear Elliptic Equations,'', Ph.D. thesis, (2009).   Google Scholar [29] P. Winkert, Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values,, Adv. Differential Equations, 15 (2010), 561.   Google Scholar [30] P. Winkert, Local $C^1(\bar \Omega)$-minimizers versus local $W^{1,p}(\Omega)$-minimizers of nonsmooth functionals, , Nonlinear Anal., 72 (2010), 4298.  doi: 10.1016/j.na.2010.02.006.  Google Scholar [31] P. Winkert, $L^\infty$-estimates for nonlinear elliptic Neumann boundary value problems, , NoDEA Nonlinear Differential Equations Appl., 17 (2010), 289.  doi: 10.1007/s00030-009-0054-5.  Google Scholar [32] P. Winkert, Sign-changing and extremal constant-sign solutions of nonlinear elliptic Neumann boundary value problems,, Bound. Value Probl., (2010).  doi: 10.1155/2010/139126.  Google Scholar [33] P. Winkert and R. Zacher, A priori bounds of solutions to elliptic equations with nonstandard growth,, Discrete Contin. Dyn. Syst. Series S, 5 (2012), 865.  doi: 10.3934/dcdss.2012.5.865.  Google Scholar [34] E. Zeidler, "Nonlinear Functional Analysis and Its Applications. III,'', Springer-Verlag, (1985).   Google Scholar [35] J.-H. Zhao and P.-H. Zhao, Existence of infinitely many weak solutions for the $p$-Laplacian with nonlinear boundary conditions, , Nonlinear Anal., 69 (2008), 1343.  doi: 10.1016/j.na.2007.06.036.  Google Scholar

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References:
 [1] E. A. M. Abreu, J. Marcos do Ó and E. S. Medeiros, Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems,, Nonlinear Anal., 60 (2005), 1443.  doi: 10.1016/j.na.2004.09.058.  Google Scholar [2] R. P. Agarwal and L. Wei, Existence of solutions to nonlinear Neumann boundary value problems with generalized $p$-Laplacian operator, , Comput. Math. Appl., 56 (2008), 530.  doi: 10.1016/j.camwa.2008.01.013.  Google Scholar [3] M. Arias, J. Campos and J.-P. Gossez, On the antimaximum principle and the Fučik spectrum for the Neumann $p$-Laplacian, , Differential Integral Equations, 13 (2000), 217.   Google Scholar [4] D. Averna and G. Bonanno, Three solutions for a Neumann boundary value problem involving the $p$-Laplacian, , Matematiche (Catania), 60 (2005), 81.   Google Scholar [5] G. Bonanno and P. Candito, Three solutions to a Neumann problem for elliptic equations involving the $p$-Laplacian, , Arch. Math. (Basel), 80 (2003), 424.  doi: 10.1007/s00013-003-0479-8.  Google Scholar [6] G. Bonanno and G. D'Aguì, On the Neumann problem for elliptic equations involving the $p$-Laplacian, , J. Math. Anal. Appl., 358 (2009), 223.  doi: 10.1016/j.jmaa.2009.04.055.  Google Scholar [7] S. Carl, Existence and comparison results for noncoercive and nonmonotone multivalued elliptic problems,, Nonlinear Anal., 65 (2006), 1532.  doi: 10.1016/j.na.2005.10.028.  Google Scholar [8] S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities,'', Springer, (2007).   Google Scholar [9] S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems,, Nonlinear Anal., 68 (2008), 2668.  doi: 10.1016/j.na.2007.02.013.  Google Scholar [10] S. Carl, and D. Motreanu, Sign-changing and extremal constant-sign solutions of nonlinear elliptic problems with supercritical nonlinearities,, Comm. Appl. Nonlinear Anal., 14 (2007), 85.   Google Scholar [11] S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian, , Abstr. Appl. Anal., 7 (2002), 613.  doi: 10.1155/S1085337502207010.  Google Scholar [12] S.-G. Deng, Positive solutions for Robin problem involving the $p(x)$-Laplacian, , J. Math. Anal. Appl., 360 (2009), 548.  doi: 10.1016/j.jmaa.2009.06.032.  Google Scholar [13] J. Fernández Bonder and J. D. Rossi, Existence results for the $p$-Laplacian with nonlinear boundary conditions, , J. Math. Anal. Appl., (2001), 195.  doi: 10.1006/jmaa.2001.7609.  Google Scholar [14] J. Fernández Bonder, Multiple positive solutions for quasilinear elliptic problems with sign-changing nonlinearities,, Abstr. Appl. Anal., 12 (2004), 1047.  doi: 10.1155/S1085337504403078.  Google Scholar [15] J. Fernández Bonder, Multiple solutions for the $p$-Laplace equation with nonlinear boundary conditions, , Electron. J. Differential Equations, 37 (2006).   Google Scholar [16] L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,'', Series in Mathematical Analysis and Applications, (2005).   Google Scholar [17] Z. Jin, Multiple solutions for a class of semilinear elliptic equations,, Proc. Amer. Math. Soc., 125 (1997), 3659.  doi: 10.1090/S0002-9939-97-04199-3.  Google Scholar [18] A. Lê, Eigenvalue problems for the $p$-Laplacian, , Nonlinear Anal., 64 (2006), 1057.  doi: 10.1016/j.na.2005.05.056.  Google Scholar [19] C. Li and S. Li, Multiple solutions and sign-changing solutions of a class of nonlinear elliptic equations with Neumann boundary condition,, J. Math. Anal. Appl., 298 (2004), 14.  doi: 10.1016/j.jmaa.2004.01.017.  Google Scholar [20] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar [21] C. Liu and Y. Zheng, Linking solutions for $p$-Laplace equations with nonlinear boundary conditions and indefinite weight, , Calc. Var. Partial Differential Equations, 41 (2011), 261.  doi: 10.1007/s00526-010-0361-z.  Google Scholar [22] S. R. Martínez and J. D. and Rossi, On the Fučik spectrum and a resonance problem for the {$p$-Laplacian with a nonlinear boundary condition, , Nonlinear Anal., 59 (2004), 813.  doi: 10.1016/j.na.2004.07.039.  Google Scholar [23] S. R. Martínez and J. D. Rossi, Weak solutions for the $p$-Laplacian with a nonlinear boundary condition at resonance, , Electron. J. Differential Equations, 27 (2003).   Google Scholar [24] D. Motreanu and P. Winkert, On the Fučik Spectrum for the $p$-Laplacian with a Robin boundary condition,, Nonlinear Anal., 74 (2011), 4671.  doi: 10.1016/j.na.2011.04.033.  Google Scholar [25] D. Motreanu and P. Winkert, The Fučik spectrum for the negative $p$-Laplacian with different boundary conditions,, Springer Optimization and Its Applications, (2012), 471.  doi: 10.1007/978-1-4614-3498-6_28.  Google Scholar [26] P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Regional Conference Series in Mathematics, (1986).   Google Scholar [27] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar [28] P. Winkert, "Comparison Principles and Multiple Solutions for Nonlinear Elliptic Equations,'', Ph.D. thesis, (2009).   Google Scholar [29] P. Winkert, Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values,, Adv. Differential Equations, 15 (2010), 561.   Google Scholar [30] P. Winkert, Local $C^1(\bar \Omega)$-minimizers versus local $W^{1,p}(\Omega)$-minimizers of nonsmooth functionals, , Nonlinear Anal., 72 (2010), 4298.  doi: 10.1016/j.na.2010.02.006.  Google Scholar [31] P. Winkert, $L^\infty$-estimates for nonlinear elliptic Neumann boundary value problems, , NoDEA Nonlinear Differential Equations Appl., 17 (2010), 289.  doi: 10.1007/s00030-009-0054-5.  Google Scholar [32] P. Winkert, Sign-changing and extremal constant-sign solutions of nonlinear elliptic Neumann boundary value problems,, Bound. Value Probl., (2010).  doi: 10.1155/2010/139126.  Google Scholar [33] P. Winkert and R. Zacher, A priori bounds of solutions to elliptic equations with nonstandard growth,, Discrete Contin. Dyn. Syst. Series S, 5 (2012), 865.  doi: 10.3934/dcdss.2012.5.865.  Google Scholar [34] E. Zeidler, "Nonlinear Functional Analysis and Its Applications. III,'', Springer-Verlag, (1985).   Google Scholar [35] J.-H. Zhao and P.-H. Zhao, Existence of infinitely many weak solutions for the $p$-Laplacian with nonlinear boundary conditions, , Nonlinear Anal., 69 (2008), 1343.  doi: 10.1016/j.na.2007.06.036.  Google Scholar
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