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

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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

Nikolaos S. Papageorgiou ^1, , Vicenţiu D. Rădulescu ^2,, and Dušan D. Repovš ^3,

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}$.

Keywords: Indefinite and unbounded potential, Robin eigenvalue problem, sublinear perturbation, superlinear perturbation, maximum principle, positive solution, minimal positive solution.

Mathematics Subject Classification: Primary: 35J20; Secondary: 35J60.

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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