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

Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu; Dušan D. Repovš

doi:10.3934/dcds.2017111

Article Contents

2017, Volume 37, Issue 5: 2589-2618. Doi: 10.3934/dcds.2017111

This issue Previous Article Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves Next Article Traveling fronts bifurcating from stable layers in the presence of conservation laws

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
* Corresponding author: Vicenţiu D. Rădulescu

Received: April 2016

Revised: December 2016

Published: May 2017

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.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[2]	A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381.
[3]	H. Brezis and L. Nirenberg, $ H^{1}$ versus $ C^{1}$ local minimizers, C.R. Acad. Sci. Paris, Sér. Ⅰ, 317 (1993), 465-472.
[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.
[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.
[6]	L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.