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

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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

Abstract
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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

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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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