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

Double resonance for Robin problems with indefinite and unbounded potential

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.

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'AguìS. 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. MotreanuV. 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'AguìS. 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. MotreanuV. 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

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