American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity
  • CPAA Home
  • This Issue
  • Next Article
    Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data
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, J. Funct. Anal., 14 (1973), 349-381.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  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, J. Eur. Math. Soc., 8 (2006), 269-288. 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, J. Differential Equations, 198 (2004), 91-128. 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, J. Funct. Anal., 225 (2005), 352-370. doi: 10.1016/j.jfa.2005.04.005.  Google Scholar

[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.  Google Scholar

[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.  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, American Mathematical Society, Providence, RI, 1986.  Google Scholar

[13]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  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, J. Eur. Math. Soc., 8 (2006), 269-288. 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, J. Differential Equations, 198 (2004), 91-128. 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, J. Funct. Anal., 225 (2005), 352-370. doi: 10.1016/j.jfa.2005.04.005.  Google Scholar

[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.  Google Scholar

[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.  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, American Mathematical Society, Providence, RI, 1986.  Google Scholar

[13]

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

[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.  Google Scholar

[1]

D. Motreanu, Donal O'Regan, Nikolaos S. Papageorgiou. A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1791-1816. doi: 10.3934/cpaa.2011.10.1791
[2]

Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071
[3]

Xiying Sun, Qihuai Liu, Dingbian Qian, Na Zhao. Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 279-292. doi: 10.3934/cpaa.20200015
[4]

Lushun Wang, Minbo Yang, Yu Zheng. Infinitely many segregated solutions for coupled nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 6069-6102. doi: 10.3934/dcds.2019265
[5]

Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3407-3428. doi: 10.3934/dcdsb.2016104
[6]

Petru Jebelean. Infinitely many solutions for ordinary $p$-Laplacian systems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (2) : 267-275. doi: 10.3934/cpaa.2008.7.267
[7]

Weiwei Ao, Juncheng Wei, Wen Yang. Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5561-5601. doi: 10.3934/dcds.2017242
[8]

Weiwei Ao, Liping Wang, Wei Yao. Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials. Communications on Pure & Applied Analysis, 2016, 15 (3) : 965-989. doi: 10.3934/cpaa.2016.15.965
[9]

Wei Long, Shuangjie Peng, Jing Yang. Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 917-939. doi: 10.3934/dcds.2016.36.917
[10]

Weiming Liu, Chunhua Wang. Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7081-7115. doi: 10.3934/dcds.2016109
[11]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094
[12]

Andrzej Szulkin, Shoyeb Waliullah. Infinitely many solutions for some singular elliptic problems. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 321-333. doi: 10.3934/dcds.2013.33.321
[13]

Joseph Iaia. Existence of infinitely many solutions for semilinear problems on exterior domains. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4269-4284. doi: 10.3934/cpaa.2020193
[14]

Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure & Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003
[15]

Liang Zhang, X. H. Tang, Yi Chen. Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 823-842. doi: 10.3934/cpaa.2017039
[16]

Alberto Boscaggin, Anna Capietto. Infinitely many solutions to superquadratic planar Dirac-type systems. Conference Publications, 2009, 2009 (Special) : 72-81. doi: 10.3934/proc.2009.2009.72
[17]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51
[18]

Ziheng Zhang, Rong Yuan. Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials. Communications on Pure & Applied Analysis, 2014, 13 (2) : 623-634. doi: 10.3934/cpaa.2014.13.623
[19]

Dušan D. Repovš. Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 401-411. doi: 10.3934/dcdss.2019026
[20]

Yinbin Deng, Shuangjie Peng, Li Wang. Infinitely many radial solutions to elliptic systems involving critical exponents. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 461-475. doi: 10.3934/dcds.2014.34.461

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences