Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
See all formats and pricing
More options …
Volume 30, Issue 3

Issues

Positive solutions for nonlinear nonhomogeneous parametric Robin problems

Nikolaos S. Papageorgiou / Vicenţiu D. RădulescuORCID iD: http://orcid.org/0000-0003-4615-5537
  • Corresponding author
  • Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia; and Department of Mathematics, University of Craiova, Street A. I. Cuza 13, 200585 Craiova, Romania
  • orcid.org/0000-0003-4615-5537
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Dušan D. Repovš
Published Online: 2017-08-15 | DOI: https://doi.org/10.1515/forum-2017-0124

Abstract

We study a parametric Robin problem driven by a nonlinear nonhomogeneous differential operator and with a superlinear Carathéodory reaction term. We prove a bifurcation-type theorem for small values of the parameter. Also, we show that as the parameter λ>0 approaches zero, we can find positive solutions with arbitrarily big and arbitrarily small Sobolev norm. Finally, we show that for every admissible parameter value, there is a smallest positive solution uλ* of the problem, and we investigate the properties of the map λuλ*.

Keywords: Robin boundary condition; nonlinear nonhomogeneous differential operator; nonlinear regularity; nonlinear maximum principle; bifurcation-type result; extremal positive solution

MSC 2010: 35J20; 35J60

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), no. 915, 1–70. Google Scholar

  • [2]

    S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nodal solutions for (p,2)-equations, Trans. Amer. Math. Soc. 367 (2015), no. 10, 7343–7372. Google Scholar

  • [3]

    S. Aizicovici, N. S. Papageorgiou and V. Staicu, Positive solutions for parametric nonlinear nonhomogeneous Robin problems, Funkcial. Ekvac., to appear. Google Scholar

  • [4]

    A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), no. 2, 519–543. CrossrefGoogle Scholar

  • [5]

    A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. CrossrefGoogle Scholar

  • [6]

    D. Arcoya and D. Ruiz, The Ambrosetti–Prodi problem for the p-Laplacian operator, Comm. Partial Differential Equations 31 (2006), no. 4–6, 849–865. CrossrefGoogle Scholar

  • [7]

    V. Benci, P. D’Avenia, D. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick’s problem and infinitely many solutions, Arch. Ration. Mech. Anal. 154 (2000), no. 4, 297–324. CrossrefGoogle Scholar

  • [8]

    E. Casas and L. A. Fernández, A Green’s formula for quasilinear elliptic operators, J. Math. Anal. Appl. 142 (1989), no. 1, 62–73. CrossrefGoogle Scholar

  • [9]

    L. Cherfils and Y. Ilyasov, On the stationary solutions of generalized reaction-diffusion equations with p&q Laplacian, Commun. Pure Appl. Anal. 4 (2005), 9–22. Google Scholar

  • [10]

    S. Cingolani and M. Degiovanni, Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity, Comm. Partial Differential Equations 30 (2005), no. 7–9, 1191–1203. CrossrefGoogle Scholar

  • [11]

    J. I. Díaz and J. E. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 12, 521–524. Google Scholar

  • [12]

    M. E. Filippakis and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations 245 (2008), no. 7, 1883–1922. CrossrefWeb of ScienceGoogle Scholar

  • [13]

    M. Fuchs and V. Osmolovski, Variational integrals on Orlicz–Sobolev spaces, Z. Anal. Anwend. 17 (1998), no. 2, 393–415. CrossrefGoogle Scholar

  • [14]

    J. P. García Azorero, I. Peral Alonso and J. J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2 (2000), no. 3, 385–404. CrossrefGoogle Scholar

  • [15]

    L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Ser. Math. Anal. Appl. 9, Chapman & Hall/CRC, Boca Raton, 2006. Google Scholar

  • [16]

    L. Gasiński and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann p-Laplacian-type equations, Adv. Nonlinear Stud. 8 (2008), no. 4, 843–870. Google Scholar

  • [17]

    L. Gasiński and N. S. Papageorgiou, Bifurcation-type results for nonlinear parametric elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), no. 3, 595–623. CrossrefGoogle Scholar

  • [18]

    M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), no. 8, 879–902. CrossrefGoogle Scholar

  • [19]

    Z. Guo and Z. Zhang, W1,p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl. 286 (2003), no. 1, 32–50. Google Scholar

  • [20]

    S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I: Theory, Math. Appl. 419, Kluwer Academic Publishers, Dordrecht, 1997. Google Scholar

  • [21]

    S. Hu and N. S. Papageorgiou, Multiplicity of solutions for parametric p-Laplacian equations with nonlinearity concave near the origin, Tohoku Math. J. (2) 62 (2010), no. 1, 137–162. CrossrefWeb of ScienceGoogle Scholar

  • [22]

    S. Hu and N. S. Papageorgiou, Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Commun. Pure Appl. Anal. 10 (2011), no. 4, 1055–1078. CrossrefWeb of ScienceGoogle Scholar

  • [23]

    N. Kenmochi, Pseudomonotone operators and nonlinear elliptic boundary value problems, J. Math. Soc. Japan 27 (1975), 121–149. CrossrefGoogle Scholar

  • [24]

    G. Li and C. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti–Rabinowitz condition, Nonlinear Anal. 72 (2010), no. 12, 4602–4613. Web of ScienceCrossrefGoogle Scholar

  • [25]

    G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), no. 2–3, 311–361. CrossrefGoogle Scholar

  • [26]

    D. Motreanu, V. V. Motreanu and N. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. Google Scholar

  • [27]

    D. Mugnai and N. S. Papageorgiou, Wang’s multiplicity result for superlinear (p,q)-equations without the Ambrosetti–Rabinowitz condition, Trans. Amer. Math. Soc. 366 (2014), no. 9, 4919–4937. Google Scholar

  • [28]

    N. S. Papageorgiou and V. D. Rădulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations 256 (2014), no. 7, 2449–2479. CrossrefWeb of ScienceGoogle Scholar

  • [29]

    N. S. Papageorgiou and V. D. Rădulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim. 69 (2014), no. 3, 393–430. Web of ScienceCrossrefGoogle Scholar

  • [30]

    N. S. Papageorgiou and V. D. Rădulescu, Bifurcation near infinity for the Robin p-Laplacian, Manuscripta Math. 148 (2015), no. 3–4, 415–433. Web of ScienceCrossrefGoogle Scholar

  • [31]

    N. S. Papageorgiou and V. D. Rădulescu, Nonlinear parametric Robin problems with combined nonlinearities, Adv. Nonlinear Stud. 15 (2015), no. 3, 715–748. Google Scholar

  • [32]

    N. S. Papageorgiou and V. D. Rădulescu, Resonant (p,2)-equations with asymmetric reaction, Anal. Appl. (Singap.) 13 (2015), no. 5, 481–506. Google Scholar

  • [33]

    N. S. Papageorgiou and V. D. Rădulescu, Solutions with sign information for nonlinear nonhomogeneous elliptic equations, Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 575–600. Web of ScienceCrossrefGoogle Scholar

  • [34]

    N. S. Papageorgiou and V. D. Rădulescu, Infinitely many nodal solutions for nonlinear nonhomogeneous Robin problems, Adv. Nonlinear Stud. 16 (2016), no. 2, 287–299. Web of ScienceGoogle Scholar

  • [35]

    N. S. Papageorgiou and V. D. Rădulescu, Noncoercive resonant (p,2)-equations, Appl. Math. Optim. (2016), 10.1007/s00245-016-9363-3. Web of ScienceGoogle Scholar

  • [36]

    N. S. Papageorgiou and V. D. Rădulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud. 16 (2016), no. 4, 737–764. Web of ScienceGoogle Scholar

  • [37]

    N. S. Papageorgiou and V. D. Rădulescu, Multiplicity theorems for nonlinear nonhomogeneous Robin problems, Rev. Mat. Iberoam. 33 (2017), no. 1, 251–289. CrossrefWeb of ScienceGoogle Scholar

  • [38]

    N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, On a class of parametric (p,2)-equations, Appl. Math. Optim. 75 (2017), no. 2, 193–228. Web of ScienceGoogle Scholar

  • [39]

    N. S. Papageorgiou and G. Smyrlis, A bifurcation-type theorem for singular nonlinear elliptic equations, Methods Appl. Anal. 22 (2015), no. 2, 147–170. Web of ScienceGoogle Scholar

  • [40]

    N. S. Papageorgiou and P. Winkert, Resonant (p,2)-equations with concave terms, Appl. Anal. 94 (2015), no. 2, 342–360. Web of ScienceGoogle Scholar

  • [41]

    P. Pucci and J. Serrin, The Maximum Principle, Progr. Nonlinear Differential Equations Appl. 73, Birkhäuser, Basel, 2007. Google Scholar

  • [42]

    M. Sun, M. Zhang and J. Su, Critical groups at zero and multiple solutions for a quasilinear elliptic equation, J. Math. Anal. Appl. 428 (2015), no. 1, 696–712. CrossrefWeb of ScienceGoogle Scholar

About the article


Received: 2017-06-12

Revised: 2017-07-13

Published Online: 2017-08-15

Published in Print: 2018-05-01


Funding Source: Javna Agencija za Raziskovalno Dejavnost RS

Award identifier / Grant number: P1-0292

Award identifier / Grant number: J1-8131

Award identifier / Grant number: J1-7025

This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025. V. D. Rădulescu was also supported by a grant of Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III.


Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 553–580, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0124.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Nikolaos S. Papageorgiou, Calogero Vetro, and Francesca Vetro
Revista Matemática Complutense, 2019
[2]
Nikolaos S. Papageorgiou and Andrea Scapellato
Advances in Nonlinear Analysis, 2019, Volume 9, Number 1, Page 449

Comments (0)

Please log in or register to comment.
Log in