Abstract
We consider a parametric Robin problem driven by the p-Laplacian and with a Carathéodory reaction. Our hypotheses on the reaction incorporate a special case p-logistic equations with a superdiffusive reaction. Using variational methods coupled with suitable truncation, perturbation and comparison techniques, we prove a bifurcation near infinity result.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Amer. Math. Soc. 196(915), vi+70 (2008)
Ambrosetti A., Rabinowitz P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Brock F., Itturiaga L., Ubilla P.: A multiplicity result for the p-Laplacian involving a parameter. Ann. Henri Poincaré 9, 137–1386 (2008)
Cardinali T., Papageorgiou N.S., Rubbioni P.: Bifurcation phenomena for nonlinear superdiffusive Neuman equations of logistic type. Ann. Mat. Pura Appl. (4) 193, 1–21 (2014)
Casas E., Fernandez L.A.: A Green’s formula for quasilinear elliptic operators. J. Math. Anal. Appl. 142, 62–73 (1989)
Denkowski Z., Migorski S., Papageorgiou N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic Publishers, Boston (2003)
DiBenedetto E.: \({C^{1+\alpha}}\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850 (1983)
Filippakis M., O’Regan D., Papageorgiou N.S.: Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: the superdiffusive case. Commun. Pure Appl. Anal. 9, 1507–1527 (2010)
Gasinski L., Papageorgiou N.S.: Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton (2006)
Gasinski L., Papageorgiou N.S.: Bifurcation-type results for nonlinear parametric elliptic equations. Proc. R. Soc. Edinb. Sect. A 142, 595–623 (2012)
Goldberg S.: Unbounded Linear Operators: Theory and Applications. McGraw-Hill, New York (1966)
Kenmochi N.: Pseudomonotone operators and nonlinear elliptic boundary value problems. J. Math. Soc. Jpn. 27, 121–149 (1975)
Lieberman G.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)
Papageorgiou N.S., Rădulescu V.D.: Multiple solutions with precise sign information for nonlinear parametric Robin problems. J. Differ. Equ. 256, 2449–2479 (2014)
Papageorgiou N.S., Rădulescu V.D.: Positive solutions for nonlinear nonhomogeneous Neuman equations of superdiffusive type. J. Fixed Point Theory Appl. 15, 519–535 (2014)
Papageorgiou N.S., Rădulescu V.D.: Bifurcation near the origin for the Robin problem with concaveconvex nonlinearities. C. R. Acad. Sci. Paris 352, 627–632 (2014)
Rădulescu V.D., Repovš D.: Combined effects in nonlinear problems arising in the study of anisotropic continuous media. Nonlinear Anal. 75, 1524–1530 (2012)
Schaefer H.H.: Banach Lattices and Positive Operators. Springer, New York (1974)
Takeuchi S.: Positive solutions of a degenerate elliptic equation with a logistic reaction. Proc. Am. Math. Soc. 129, 433–441 (2001)
Takeuchi S.: Multiplicity results for a degenerate elliptic equation with a logistic reaction. J. Differ. Equ. 173, 138–144 (2001)
Tolksdorf P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)
Winkert P.: \({L^{\infty}}\) estimates for nonlinear elliptic Neumann boundary value problems. NoDEA Nonlinear Differ. Equ. Appl. 17, 289–302 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Papageorgiou, N.S., Rădulescu, V.D. Bifurcation near infinity for the Robin p-Laplacian. manuscripta math. 148, 415–433 (2015). https://doi.org/10.1007/s00229-015-0754-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-015-0754-6