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Positive solutions for nonlinear nonhomogeneous parametric Robin problems

Nikolaos S. Papageorgiou
/ Vicenţiu D. Rădulescu
• 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:
/ 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 $\lambda >0$ 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 ${u}_{\lambda }^{*}$ of the problem, and we investigate the properties of the map $\lambda ↦{u}_{\lambda }^{*}$.

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 $\left(p,2\right)$-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.

• [5]

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

• [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.

• [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.

• [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.

• [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.

• [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.

• [13]

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

• [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.

• [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.

• [18]

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

• [19]

Z. Guo and Z. Zhang, ${W}^{1,p}$ versus ${C}^{1}$ 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.

• [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.

• [23]

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

• [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.

• [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.

• [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 $\left(p,q\right)$-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.

• [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.

• [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.

• [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 $\left(p,2\right)$-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.

• [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.

• [35]

N. S. Papageorgiou and V. D. Rădulescu, Noncoercive resonant $\left(p,2\right)$-equations, Appl. Math. Optim. (2016), 10.1007/s00245-016-9363-3.

• [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.

• [37]

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

• [38]

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

• [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.

• [40]

N. S. Papageorgiou and P. Winkert, Resonant $\left(p,2\right)$-equations with concave terms, Appl. Anal. 94 (2015), no. 2, 342–360.

• [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.

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,

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