# American Institute of Mathematical Sciences

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.

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##### 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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