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July  2014, 13(4): 1593-1612. doi: 10.3934/cpaa.2014.13.1593

Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions

Ryuji Kajikiya 1, and  Daisuke Naimen 2,

1. 

Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan
2. 

Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto Sumiyoshi-ku, Osaka-shi, Osaka, 558-8585, Japan

Received  October 2013 Revised  January 2014 Published  February 2014

In this paper, we study semilinear elliptic equations with nonlinear Neumann boundary conditions. We prove the existence of a sequence of solutions converging to zero if the nonlinear term is locally sublinear and the existence of a sequence of solutions diverging to infinity if the nonlinear term is locally superlinear.
Keywords: infinitely many solutions, variational method., Superlinear, sublinear, nonlinear Neumann.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J25, 35J6.
Citation: Ryuji Kajikiya, Daisuke Naimen. Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1593-1612. doi: 10.3934/cpaa.2014.13.1593
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349.   Google Scholar

[2]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems,, \emph{J. Funct. Anal.}, 122 (1994), 519.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[3]

T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities,, \emph{Proc. Amer. Math. Soc.}, 123 (1995), 3555.  doi: 10.2307/2161107.  Google Scholar

[4]

D. C. Clark, A variant of the Lusternik-Schnirelman theory,, \emph{Indiana Univ. Math. J.}, 22 (1972), 65.   Google Scholar

[5]

D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems,, \emph{J. Funct. Anal.}, 199 (2003), 452.  doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[6]

D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity,, \emph{J. Eur. Math. Soc.}, 8 (2006), 269.  doi: 10.4171/JEMS/52.  Google Scholar

[7]

J. Garcia-Azorero, I. Peral and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition,, \emph{J. Differential Equations}, 198 (2004), 91.  doi: 10.1016/S0022-0396(03)00068-8.  Google Scholar

[8]

R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations,, \emph{J. Funct. Anal.}, 225 (2005), 352.  doi: 10.1016/j.jfa.2005.04.005.  Google Scholar

[9]

R. Kajikiya, Superlinear elliptic equations with singular coefficients on the boundary,, \emph{Nonlinear Analysis, 73 (2010), 2117.  doi: 10.1016/j.na.2010.05.039.  Google Scholar

[10]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations,, translated by Scripta Technica, (1968).   Google Scholar

[11]

D. Naimen, Existence of infinitely many solutions for nonlinear Neumann problems with indefinite coefficients,, Submitted for publications., ().   Google Scholar

[12]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Regional Conference Series in Mathematics Vol. 65, (1986).   Google Scholar

[13]

M. Struwe, Variational Methods,, 2$^{nd}$ edition, (1996).   Google Scholar

[14]

Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin,, \emph{Nonlinear Differential Equations Appl.}, 8 (2001), 15.  doi: 10.1007/PL00001436.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349.   Google Scholar

[2]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems,, \emph{J. Funct. Anal.}, 122 (1994), 519.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[3]

T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities,, \emph{Proc. Amer. Math. Soc.}, 123 (1995), 3555.  doi: 10.2307/2161107.  Google Scholar

[4]

D. C. Clark, A variant of the Lusternik-Schnirelman theory,, \emph{Indiana Univ. Math. J.}, 22 (1972), 65.   Google Scholar

[5]

D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems,, \emph{J. Funct. Anal.}, 199 (2003), 452.  doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[6]

D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity,, \emph{J. Eur. Math. Soc.}, 8 (2006), 269.  doi: 10.4171/JEMS/52.  Google Scholar

[7]

J. Garcia-Azorero, I. Peral and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition,, \emph{J. Differential Equations}, 198 (2004), 91.  doi: 10.1016/S0022-0396(03)00068-8.  Google Scholar

[8]

R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations,, \emph{J. Funct. Anal.}, 225 (2005), 352.  doi: 10.1016/j.jfa.2005.04.005.  Google Scholar

[9]

R. Kajikiya, Superlinear elliptic equations with singular coefficients on the boundary,, \emph{Nonlinear Analysis, 73 (2010), 2117.  doi: 10.1016/j.na.2010.05.039.  Google Scholar

[10]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations,, translated by Scripta Technica, (1968).   Google Scholar

[11]

D. Naimen, Existence of infinitely many solutions for nonlinear Neumann problems with indefinite coefficients,, Submitted for publications., ().   Google Scholar

[12]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Regional Conference Series in Mathematics Vol. 65, (1986).   Google Scholar

[13]

M. Struwe, Variational Methods,, 2$^{nd}$ edition, (1996).   Google Scholar

[14]

Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin,, \emph{Nonlinear Differential Equations Appl.}, 8 (2001), 15.  doi: 10.1007/PL00001436.  Google Scholar

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