October  2015, 35(10): 5003-5036. doi: 10.3934/dcds.2015.35.5003

Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities

1. 

Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

2. 

“Simion Stoilow” Mathematics Institute of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

Received  September 2014 Revised  January 2015 Published  April 2015

In this paper we deal with Robin and Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing nonlinearities (concave-convex nonlinearities). For the Robin problem and without employing the Ambrosetti-Rabinowitz condition, we prove a bifurcation theorem for the positive solutions for small values of the parameter $\lambda>0$. For the Neumann problem with a different geometry and using the Ambrosetti-Rabinowitz condition we prove bifurcation for large values of $\lambda>0$.
Citation: Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5003-5036. doi: 10.3934/dcds.2015.35.5003
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, H. Brezis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[3]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[4]

F. Cîrstea, M. Ghergu and V. D. Rădulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type, J. Math. Pures Appl., 84 (2005), 493-508. doi: 10.1016/j.matpur.2004.09.005.

[5]

J. I. Diaz and J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C.R. Acad. Sci. Paris, 305 (1987), 521-524.

[6]

N. Dunford and J. Schwartz, Linear Operators I, Wiley-Interscience, New York, 1958.

[7]

D. G. de Figueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-286. doi: 10.4171/JEMS/52.

[8]

D. G. de Figueiredo, J.-P. Gossez and P. Ubilla, Local "superlinearity" and "sublinearity" for the $p$-Laplacian, J. Funct. Anal., 257 (2009), 721-752. doi: 10.1016/j.jfa.2009.04.001.

[9]

M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nonzero solutions with constant sign for a $p$-Laplacian equation, Discrete Cont. Dyn. Systems, 24 (2009), 405-440. doi: 10.3934/dcds.2009.24.405.

[10]

J. Garcia Azero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., 2 (2000), 385-404. doi: 10.1142/S0219199700000190.

[11]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman, Hall/CRC, Boca Raton, Fl., 2006.

[12]

L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian-type equations, Adv. Nonlinear Studies, 8 (2008), 843-870.

[13]

L. Gasinski and N. S. Papageorgiou, Bifurcation-type results for nonlinear parametric elliptic equations, Proc. Royal. Soc. Edinburgh, Sect. A, 142 (2012), 595-623. doi: 10.1017/S0308210511000126.

[14]

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), 32-50. doi: 10.1016/S0022-247X(03)00282-8.

[15]

S. Hu and N. S. Papageorgiou, Multiplicity of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin, Tôhoku Math. J., 62 (2010), 137-162. doi: 10.2748/tmj/1270041030.

[16]

A. Kristaly and G. Moroşanu, New competition phenomena in Dirichlet problems, J. Math. Pures Appl., 94 (2010), 555-570. doi: 10.1016/j.matpur.2010.03.005.

[17]

G. Lieberman, The natural generalization of the conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Commun. Partial Diff. Equations, 16 (1991), 311-361. doi: 10.1080/03605309108820761.

[18]

N. S. Papageorgiou and V. D. Rădulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim., 69 (2014), 393-430. doi: 10.1007/s00245-013-9227-z.

[19]

N. S. Papageorgiou and V. D. Rădulescu, Solutions with sign information for nonlinear nonhomogeneous elliptic equations, Topol. Methods Nonlin. Anal. to appear.

[20]

N. S. Papageorgiou and V. D. Rădulescu, 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.

[21]

P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.

[22]

V. D. Rădulescu and D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530. doi: 10.1016/j.na.2011.01.037.

[23]

P. Winkert, $L^{\infty}$ estimates for nonlinear elliptic Neumann boundary value problems, Nonlinear Diff. Equ. Appl. (NoDEA), 17 (2010), 289-302. doi: 10.1007/s00030-009-0054-5.

show all references

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, H. Brezis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[3]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[4]

F. Cîrstea, M. Ghergu and V. D. Rădulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type, J. Math. Pures Appl., 84 (2005), 493-508. doi: 10.1016/j.matpur.2004.09.005.

[5]

J. I. Diaz and J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C.R. Acad. Sci. Paris, 305 (1987), 521-524.

[6]

N. Dunford and J. Schwartz, Linear Operators I, Wiley-Interscience, New York, 1958.

[7]

D. G. de Figueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-286. doi: 10.4171/JEMS/52.

[8]

D. G. de Figueiredo, J.-P. Gossez and P. Ubilla, Local "superlinearity" and "sublinearity" for the $p$-Laplacian, J. Funct. Anal., 257 (2009), 721-752. doi: 10.1016/j.jfa.2009.04.001.

[9]

M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nonzero solutions with constant sign for a $p$-Laplacian equation, Discrete Cont. Dyn. Systems, 24 (2009), 405-440. doi: 10.3934/dcds.2009.24.405.

[10]

J. Garcia Azero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., 2 (2000), 385-404. doi: 10.1142/S0219199700000190.

[11]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman, Hall/CRC, Boca Raton, Fl., 2006.

[12]

L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian-type equations, Adv. Nonlinear Studies, 8 (2008), 843-870.

[13]

L. Gasinski and N. S. Papageorgiou, Bifurcation-type results for nonlinear parametric elliptic equations, Proc. Royal. Soc. Edinburgh, Sect. A, 142 (2012), 595-623. doi: 10.1017/S0308210511000126.

[14]

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), 32-50. doi: 10.1016/S0022-247X(03)00282-8.

[15]

S. Hu and N. S. Papageorgiou, Multiplicity of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin, Tôhoku Math. J., 62 (2010), 137-162. doi: 10.2748/tmj/1270041030.

[16]

A. Kristaly and G. Moroşanu, New competition phenomena in Dirichlet problems, J. Math. Pures Appl., 94 (2010), 555-570. doi: 10.1016/j.matpur.2010.03.005.

[17]

G. Lieberman, The natural generalization of the conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Commun. Partial Diff. Equations, 16 (1991), 311-361. doi: 10.1080/03605309108820761.

[18]

N. S. Papageorgiou and V. D. Rădulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim., 69 (2014), 393-430. doi: 10.1007/s00245-013-9227-z.

[19]

N. S. Papageorgiou and V. D. Rădulescu, Solutions with sign information for nonlinear nonhomogeneous elliptic equations, Topol. Methods Nonlin. Anal. to appear.

[20]

N. S. Papageorgiou and V. D. Rădulescu, 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.

[21]

P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.

[22]

V. D. Rădulescu and D. Repovš, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530. doi: 10.1016/j.na.2011.01.037.

[23]

P. Winkert, $L^{\infty}$ estimates for nonlinear elliptic Neumann boundary value problems, Nonlinear Diff. Equ. Appl. (NoDEA), 17 (2010), 289-302. doi: 10.1007/s00030-009-0054-5.

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