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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 |
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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