American Institute of Mathematical Sciences

Advanced
April  2018, 11(2): 323-344. doi: 10.3934/dcdss.2018018

Double resonance for Robin problems with indefinite and unbounded potential

Nikolaos S. Papageorgiou 1, and  Patrick Winkert 2,

1. 

National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
2. 

Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany

Received  December 2016 Revised  April 2017 Published  January 2018

We study a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential term. The nonlinearity $f(x, s)$ is a Carathéodory function which is asymptotically linear as $ s\to ± ∞$ and resonant. In fact we assume double resonance with respect to any nonprincipal, nonnegative spectral interval $ \left[ \hat{λ}_k, \hat{λ}_{k+1}\right]$. Applying variational tools along with suitable truncation and perturbation techniques as well as Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of constant sign.

Keywords: Indefinite potential, Robin boundary condition, regularity theory, critical groups, multiple nontrivial solutions, double resonance.
Mathematics Subject Classification: Primary: 35J20, 35J60, 58E05.
Citation: Nikolaos S. Papageorgiou, Patrick Winkert. Double resonance for Robin problems with indefinite and unbounded potential. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 323-344. doi: 10.3934/dcdss.2018018
References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints Mem. Amer. Math. Soc. 196 (2008), ⅵ+70 pp. doi: 10.1090/memo/0915.  Google Scholar

[2]

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

[3]

H. Berestycki and D. G. de Figueiredo, Double resonance in semilinear elliptic problems, Comm. Partial Differential Equations, 6 (1981), 91-120.  doi: 10.1080/03605308108820172.  Google Scholar

[4]

N. P. Các, On an elliptic boundary value problem at double resonance, J. Math. Anal. Appl., 132 (1988), 473-483.  doi: 10.1016/0022-247X(88)90075-3.  Google Scholar

[5] K.-C. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005.   Google Scholar
[6]

G. D'Aguì, S. A. Marano and N. S. Papageorgiou, Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction, J. Math. Anal. Appl., 433 (2016), 1821-1845.  doi: 10.1016/j.jmaa.2015.08.065.  Google Scholar

[7] L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.   Google Scholar
[8]

L. Gasiński and N. S. Papageorgiou, Neumann problems resonant at zero and infinity, Ann. Mat. Pura Appl. (4), 191 (2012), 395-430.  doi: 10.1007/s10231-011-0188-z.  Google Scholar

[9] L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2: Nonlinear Analysis, Springer, Heidelberg, 2016.  doi: 10.1007/978-3-319-27817-9.  Google Scholar
[10]

S. Li and J. Q. Liu, Computations of critical groups at degenerate critical point and applications to nonlinear differential equations with resonance, Houston J. Math., 25 (1999), 563-582.   Google Scholar

[11]

Z. Liang and J. Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158.  doi: 10.1016/j.jmaa.2008.12.053.  Google Scholar

[12] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.  doi: 10.1007/978-1-4614-9323-5.  Google Scholar
[13]

N. S. Papageorgiou and V. D. Rădulescu, Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity, Contemp. Math. , 595, Amer. Math. Soc. , Providence, RI, (2013), 293-315. doi: 10.1090/conm/595/11801.  Google Scholar

[14]

N. S. Papageorgiou and V. D. Rădulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.  doi: 10.1016/j.jde.2014.01.010.  Google Scholar

[15]

N. S. Papageorgiou and V. D. Rădulescu, Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential, Trans. Amer. Math. Soc., 367 (2015), 8723-8756.  doi: 10.1090/S0002-9947-2014-06518-5.  Google Scholar

[16]

S. Robinson, Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains, Nonlinear Anal., 21 (1993), 407-424.  doi: 10.1016/0362-546X(93)90125-C.  Google Scholar

[17]

J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895.  doi: 10.1016/S0362-546X(00)00221-2.  Google Scholar

[18]

X. J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310.  doi: 10.1016/0022-0396(91)90014-Z.  Google Scholar

[19]

W. Zou, Multiple solutions for elliptic equations with resonance, Nonlinear Anal., 48 (2002), 363-376.  doi: 10.1016/S0362-546X(00)00190-5.  Google Scholar

show all references

References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints Mem. Amer. Math. Soc. 196 (2008), ⅵ+70 pp. doi: 10.1090/memo/0915.  Google Scholar

[2]

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

[3]

H. Berestycki and D. G. de Figueiredo, Double resonance in semilinear elliptic problems, Comm. Partial Differential Equations, 6 (1981), 91-120.  doi: 10.1080/03605308108820172.  Google Scholar

[4]

N. P. Các, On an elliptic boundary value problem at double resonance, J. Math. Anal. Appl., 132 (1988), 473-483.  doi: 10.1016/0022-247X(88)90075-3.  Google Scholar

[5] K.-C. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005.   Google Scholar
[6]

G. D'Aguì, S. A. Marano and N. S. Papageorgiou, Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction, J. Math. Anal. Appl., 433 (2016), 1821-1845.  doi: 10.1016/j.jmaa.2015.08.065.  Google Scholar

[7] L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.   Google Scholar
[8]

L. Gasiński and N. S. Papageorgiou, Neumann problems resonant at zero and infinity, Ann. Mat. Pura Appl. (4), 191 (2012), 395-430.  doi: 10.1007/s10231-011-0188-z.  Google Scholar

[9] L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2: Nonlinear Analysis, Springer, Heidelberg, 2016.  doi: 10.1007/978-3-319-27817-9.  Google Scholar
[10]

S. Li and J. Q. Liu, Computations of critical groups at degenerate critical point and applications to nonlinear differential equations with resonance, Houston J. Math., 25 (1999), 563-582.   Google Scholar

[11]

Z. Liang and J. Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158.  doi: 10.1016/j.jmaa.2008.12.053.  Google Scholar

[12] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.  doi: 10.1007/978-1-4614-9323-5.  Google Scholar
[13]

N. S. Papageorgiou and V. D. Rădulescu, Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity, Contemp. Math. , 595, Amer. Math. Soc. , Providence, RI, (2013), 293-315. doi: 10.1090/conm/595/11801.  Google Scholar

[14]

N. S. Papageorgiou and V. D. Rădulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.  doi: 10.1016/j.jde.2014.01.010.  Google Scholar

[15]

N. S. Papageorgiou and V. D. Rădulescu, Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential, Trans. Amer. Math. Soc., 367 (2015), 8723-8756.  doi: 10.1090/S0002-9947-2014-06518-5.  Google Scholar

[16]

S. Robinson, Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains, Nonlinear Anal., 21 (1993), 407-424.  doi: 10.1016/0362-546X(93)90125-C.  Google Scholar

[17]

J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895.  doi: 10.1016/S0362-546X(00)00221-2.  Google Scholar

[18]

X. J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310.  doi: 10.1016/0022-0396(91)90014-Z.  Google Scholar

[19]

W. Zou, Multiple solutions for elliptic equations with resonance, Nonlinear Anal., 48 (2002), 363-376.  doi: 10.1016/S0362-546X(00)00190-5.  Google Scholar

[1]

Claudianor O. Alves, Giovany M. Figueiredo, Riccardo Molle. Multiple positive bound state solutions for a critical Choquard equation. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021061
[2]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[3]

Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021058
[4]

Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, , () : -. doi: 10.3934/era.2021022
[5]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021035
[6]

Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020
[7]

Francesca Bucci. Improved boundary regularity for a Stokes-Lamé system. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021018
[8]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[9]

Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021058
[10]

Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030
[11]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[12]

Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237
[13]

Guanwei Chen, Martin Schechter. Multiple solutions for Schrödinger lattice systems with asymptotically linear terms and perturbed terms. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021124
[14]

Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021036
[15]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[16]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[17]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[18]

Yu Yang, Jinling Zhou, Cheng-Hsiung Hsu. Critical traveling wave solutions for a vaccination model with general incidence. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021087
[19]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044
[20]

Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021041

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (67)
  • HTML views (175)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences