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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 and 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, J. Funct. Anal., 14 (1973), 349-381.

[2]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[3]

T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), 3555-3561. doi: 10.2307/2161107.

[4]

D. C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22 (1972), 65-74.

[5]

D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467. doi: 10.1016/S0022-1236(02)00060-5.

[6]

D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-288. doi: 10.4171/JEMS/52.

[7]

J. Garcia-Azorero, I. Peral and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition, J. Differential Equations, 198 (2004), 91-128. doi: 10.1016/S0022-0396(03)00068-8.

[8]

R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370. doi: 10.1016/j.jfa.2005.04.005.

[9]

R. Kajikiya, Superlinear elliptic equations with singular coefficients on the boundary, Nonlinear Analysis, T.M.A., 73 (2010), 2117-2131. doi: 10.1016/j.na.2010.05.039.

[10]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, translated by Scripta Technica, Inc. Mathematics in Science and Engineering, Vol. 46, Academic Press, 1968.

[11]

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

[12]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics Vol. 65, American Mathematical Society, Providence, RI, 1986.

[13]

M. Struwe, Variational Methods, 2nd edition, Springer, Berlin, 1996.

[14]

Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlinear Differential Equations Appl., 8 (2001), 15-33. doi: 10.1007/PL00001436.

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.

[2]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[3]

T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), 3555-3561. doi: 10.2307/2161107.

[4]

D. C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22 (1972), 65-74.

[5]

D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467. doi: 10.1016/S0022-1236(02)00060-5.

[6]

D. G. DeFigueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-288. doi: 10.4171/JEMS/52.

[7]

J. Garcia-Azorero, I. Peral and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition, J. Differential Equations, 198 (2004), 91-128. doi: 10.1016/S0022-0396(03)00068-8.

[8]

R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370. doi: 10.1016/j.jfa.2005.04.005.

[9]

R. Kajikiya, Superlinear elliptic equations with singular coefficients on the boundary, Nonlinear Analysis, T.M.A., 73 (2010), 2117-2131. doi: 10.1016/j.na.2010.05.039.

[10]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, translated by Scripta Technica, Inc. Mathematics in Science and Engineering, Vol. 46, Academic Press, 1968.

[11]

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

[12]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics Vol. 65, American Mathematical Society, Providence, RI, 1986.

[13]

M. Struwe, Variational Methods, 2nd edition, Springer, Berlin, 1996.

[14]

Z.-Q. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, Nonlinear Differential Equations Appl., 8 (2001), 15-33. doi: 10.1007/PL00001436.

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