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.

[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.

[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.

[4]

D. Averna and G. Bonanno, Three solutions for a Neumann boundary value problem involving the $p$-Laplacian, , Matematiche (Catania), 60 (2005), 81.

[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.

[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.

[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.

[8]

S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities,'', Springer, (2007).

[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.

[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.

[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.

[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.

[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.

[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.

[15]

J. Fernández Bonder, Multiple solutions for the $p$-Laplace equation with nonlinear boundary conditions, , Electron. J. Differential Equations, 37 (2006).

[16]

L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,'', Series in Mathematical Analysis and Applications, (2005).

[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.

[18]

A. Lê, Eigenvalue problems for the $p$-Laplacian, , Nonlinear Anal., 64 (2006), 1057. doi: 10.1016/j.na.2005.05.056.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[26]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Regional Conference Series in Mathematics, (1986).

[27]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191. doi: 10.1007/BF01449041.

[28]

P. Winkert, "Comparison Principles and Multiple Solutions for Nonlinear Elliptic Equations,'', Ph.D. thesis, (2009).

[29]

P. Winkert, Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values,, Adv. Differential Equations, 15 (2010), 561.

[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.

[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.

[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.

[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.

[34]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications. III,'', Springer-Verlag, (1985).

[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.

show all references

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.

[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.

[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.

[4]

D. Averna and G. Bonanno, Three solutions for a Neumann boundary value problem involving the $p$-Laplacian, , Matematiche (Catania), 60 (2005), 81.

[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.

[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.

[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.

[8]

S. Carl, V. K. Le and D. Motreanu, "Nonsmooth Variational Problems and Their Inequalities,'', Springer, (2007).

[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.

[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.

[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.

[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.

[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.

[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.

[15]

J. Fernández Bonder, Multiple solutions for the $p$-Laplace equation with nonlinear boundary conditions, , Electron. J. Differential Equations, 37 (2006).

[16]

L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,'', Series in Mathematical Analysis and Applications, (2005).

[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.

[18]

A. Lê, Eigenvalue problems for the $p$-Laplacian, , Nonlinear Anal., 64 (2006), 1057. doi: 10.1016/j.na.2005.05.056.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[26]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,'', CBMS Regional Conference Series in Mathematics, (1986).

[27]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191. doi: 10.1007/BF01449041.

[28]

P. Winkert, "Comparison Principles and Multiple Solutions for Nonlinear Elliptic Equations,'', Ph.D. thesis, (2009).

[29]

P. Winkert, Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values,, Adv. Differential Equations, 15 (2010), 561.

[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.

[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.

[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.

[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.

[34]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications. III,'', Springer-Verlag, (1985).

[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.

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