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

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. 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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