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Positive solutions of nonlinear Robin eigenvalue problems


Authors: Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu
Journal: Proc. Amer. Math. Soc. 144 (2016), 4913-4928
MSC (2010): Primary 35J66, 35J92
DOI: https://doi.org/10.1090/proc/13107
Published electronically: April 20, 2016
MathSciNet review: 3544539
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a nonlinear eigenvalue problem driven by the $ p$-
Laplacian with Robin boundary condition. Using variational methods and truncation techniques, we prove a bifurcation-type result describing the set of positive solutions as the positive parameter $ \lambda $ varies. We also produce extremal positive solutions and study their properties.


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  • [1] Sergiu Aizicovici, Nikolaos S. Papageorgiou, and Vasile Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc. 196 (2008), no. 915, vi+70. MR 2459421, https://doi.org/10.1090/memo/0915
  • [2] Tiziana Cardinali, Nikolaos S. Papageorgiou, and Paola Rubbioni, Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type, Ann. Mat. Pura Appl. (4) 193 (2014), no. 1, 1-21. MR 3158835, https://doi.org/10.1007/s10231-012-0263-0
  • [3] Wei Dong, A priori estimates and existence of positive solutions for a quasilinear elliptic equation, J. London Math. Soc. (2) 72 (2005), no. 3, 645-662. MR 2190330, https://doi.org/10.1112/S0024610705006848
  • [4] Leszek Gasiński and Nikolaos S. Papageorgiou, Nonlinear analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2168068
  • [5] Leszek Gasiński and Nikolaos S. Papageorgiou, Bifurcation-type results for nonlinear parametric elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), no. 3, 595-623. MR 2945974, https://doi.org/10.1017/S0308210511000126
  • [6] Shouchuan Hu and Nikolas S. Papageorgiou, Handbook of multivalued analysis. Vol. I, Theory, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, 1997. MR 1485775
  • [7] Gary M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203-1219. MR 969499, https://doi.org/10.1016/0362-546X(88)90053-3
  • [8] Nikolaos S. Papageorgiou and Sophia Th. Kyritsi-Yiallourou, Handbook of applied analysis, Advances in Mechanics and Mathematics, vol. 19, Springer, New York, 2009. MR 2527754
  • [9] Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations 256 (2014), no. 7, 2449-2479. MR 3160450, https://doi.org/10.1016/j.jde.2014.01.010
  • [10] Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu, Positive solutions for nonlinear nonhomogeneous Neumann equations of superdiffusive type, J. Fixed Point Theory Appl. 15 (2014), no. 2, 519-535. MR 3298011, https://doi.org/10.1007/s11784-014-0176-1
  • [11] Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu, Positive solutions for perturbations of the eigenvalue problem for the Robin $ p$-Laplacian, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 255-277. MR 3310083, https://doi.org/10.5186/aasfm.2015.4011
  • [12] Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu, Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities, Discrete Contin. Dyn. Syst. 35 (2015), no. 10, 5003-5036. MR 3392659, https://doi.org/10.3934/dcds.2015.35.5003
  • [13] Shingo Takeuchi, Positive solutions of a degenerate elliptic equation with logistic reaction, Proc. Amer. Math. Soc. 129 (2001), no. 2, 433-441 (electronic). MR 1800233, https://doi.org/10.1090/S0002-9939-00-05723-3
  • [14] Shingo Takeuchi, Multiplicity result for a degenerate elliptic equation with logistic reaction, J. Differential Equations 173 (2001), no. 1, 138-144. MR 1836247, https://doi.org/10.1006/jdeq.2000.3914
  • [15] Patrick Winkert, $ L^\infty $-estimates for nonlinear elliptic Neumann boundary value problems, NoDEA Nonlinear Differential Equations Appl. 17 (2010), no. 3, 289-302. MR 2652229, https://doi.org/10.1007/s00030-009-0054-5

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Additional Information

Nikolaos S. Papageorgiou
Affiliation: Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece
Email: npapg@math.ntua.gr

Vicenţiu D. Rădulescu
Affiliation: Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia — and — Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
Email: vicentiu.radulescu@math.cnrs.fr

DOI: https://doi.org/10.1090/proc/13107
Keywords: Superdiffusive reaction, bifurcation type theorem, extremal positive solutions, $p$-Laplacian, nonlinear regularity
Received by editor(s): May 26, 2015
Received by editor(s) in revised form: January 9, 2016, and January 21, 2016
Published electronically: April 20, 2016
Communicated by: Catherine Sulem
Article copyright: © Copyright 2016 American Mathematical Society

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