# American Institute of Mathematical Sciences

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 and 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. [2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. [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. [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. [5] K.-C. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005. [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. [7] L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] W. Zou, Multiple solutions for elliptic equations with resonance, Nonlinear Anal., 48 (2002), 363-376.  doi: 10.1016/S0362-546X(00)00190-5.

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##### 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. [2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. [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. [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. [5] K.-C. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005. [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. [7] L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] W. Zou, Multiple solutions for elliptic equations with resonance, Nonlinear Anal., 48 (2002), 363-376.  doi: 10.1016/S0362-546X(00)00190-5.
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