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May  2017, 37(5): 2589-2618. doi: 10.3934/dcds.2017111

## Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential

 1 Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece 2 Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O.Box 80203, Jeddah 21589, Saudi Arabia, Department of Mathematics, University of Craiova, Street A.I. Cuza No 13, 200585 Craiova, Romania 3 Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, Kardeljeva ploščad 16, SI-1000 Ljubljana, Slovenia

* Corresponding author: Vicenţiu D. Rădulescu

Received  April 2016 Revised  December 2016 Published  February 2017

Fund Project: This research was supported in part by the Slovenian Research Agency grants P1-0292, J1-7025 and J1-6721. V.D. Rădulescu was also supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-Ⅱ-PT-PCCA-2013-4-0614

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $λ<\widehat{λ}_{1}$ ($\widehat{λ}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $λ≥q\widehat{λ}_{1}$ there are no positive solutions. In the superlinear case, for $λ<\widehat{λ}_{1}$ we have at least two positive solutions and for $λ≥q\widehat{λ}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $\bar{u}_{λ}$ and we investigate the properties of the map $λ\mapsto\bar{u}_{λ}$.

Citation: Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2589-2618. doi: 10.3934/dcds.2017111
##### References:
 [1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Memoirs Amer. Math. Soc., 196 (2008), vi+70 pp. doi: 10.1090/memo/0915. Google Scholar [2] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381. Google Scholar [3] H. Brezis and L. Nirenberg, $H^{1}$ versus $C^{1}$ local minimizers, C.R. Acad. Sci. Paris, Sér. Ⅰ, 317 (1993), 465-472. Google Scholar [4] G. D'Agui, S. 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 [5] M. Filippakis and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations, 245 (2008), 1883-1992. doi: 10.1016/j.jde.2008.07.004. Google Scholar [6] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. Google Scholar [7] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume Ⅰ: Theory, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. doi: 10.1007/978-1-4615-4665-8_17. Google Scholar [8] S. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal., 12 (2013), 815-829. doi: 10.3934/cpaa.2013.12.815. Google Scholar [9] D. Motreanu, 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 [10] D. Mugnai, Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl. 11 (2004), no. 3,379-391, and a comment on the generalized Ambrosetti-Rabinowitz condition, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 299-301. doi: 10.1007/s00030-011-0129-y. Google Scholar [11] N. S. Papageorgiou and V. D. Rǎdulescu, 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 [12] N. S. Papageorgiou and V. D. Rǎdulescu, 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 [13] N. S. Papageorgiou and V. D. Rǎdulescu, Robin problems with indefinite, unbounded potential and reaction of arbitrary growth, Rev. Mat. Complut., 29 (2016), 91-126. doi: 10.1007/s13163-015-0181-y. Google Scholar [14] X. 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

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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, Memoirs Amer. Math. Soc., 196 (2008), vi+70 pp. doi: 10.1090/memo/0915. Google Scholar [2] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381. Google Scholar [3] H. Brezis and L. Nirenberg, $H^{1}$ versus $C^{1}$ local minimizers, C.R. Acad. Sci. Paris, Sér. Ⅰ, 317 (1993), 465-472. Google Scholar [4] G. D'Agui, S. 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 [5] M. Filippakis and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations, 245 (2008), 1883-1992. doi: 10.1016/j.jde.2008.07.004. Google Scholar [6] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. Google Scholar [7] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume Ⅰ: Theory, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. doi: 10.1007/978-1-4615-4665-8_17. Google Scholar [8] S. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal., 12 (2013), 815-829. doi: 10.3934/cpaa.2013.12.815. Google Scholar [9] D. Motreanu, 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 [10] D. Mugnai, Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl. 11 (2004), no. 3,379-391, and a comment on the generalized Ambrosetti-Rabinowitz condition, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 299-301. doi: 10.1007/s00030-011-0129-y. Google Scholar [11] N. S. Papageorgiou and V. D. Rǎdulescu, 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 [12] N. S. Papageorgiou and V. D. Rǎdulescu, 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 [13] N. S. Papageorgiou and V. D. Rǎdulescu, Robin problems with indefinite, unbounded potential and reaction of arbitrary growth, Rev. Mat. Complut., 29 (2016), 91-126. doi: 10.1007/s13163-015-0181-y. Google Scholar [14] X. 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
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