Abstract
We study an elliptic Robin problem driven by the negative Laplacian plus an indefinite and unbounded potential and with a reaction of arbitrary growth which exhibits z-dependent zeros of constant sign. We prove multiplicity theorems producing three or four nontrivial solutions, all with precise sign information. As a particular case we consider a generalized equidiffusive logistic equation with potential.
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. Am. Math. Soc. 196, 915 (2008)
Ambrosetti, A., Mancini, G.: Sharp nonuniqueness results for some nonlinear problems. Nonlinear Anal. 3, 635–645 (1979)
Ambrosetti, A., Lupo, D.: On a class of nonlinear Dirichlet problems with multiple solutions. Nonlinear Anal. 8, 1145–1150 (1984)
Brezis, H., Nirenberg, L.: \(H^1\) versus \(C^1\) local minimizers. C. R. Acad. Sci. Paris Sér. I Math. 317, 465–472 (1993)
Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston (1993)
Dunford, N., Schwartz, J.: Linear Operators I. Wiley-Interscience, New York (1958)
de Figueiredo, D., Gossez, J.P.: Strong monotonicity of eigenvalues and unique continuation. Commun. Partial Diff. Equ. 17, 339–346 (1992)
Filippakis, M., Kristaly, A., Papageorgiou, N.S.: Existence of five nonzero solutions with exact sign for a \(p\)-Laplacian operator. Discret. Contin. Dyn. Syst. 24, 405–440 (2009)
Filippakis, M., Papageorgiou, N.S.: Multiple constant sign and nodal solutions for nonlinear elliptic equations with the \(p\)-Laplacian. J. Differ. Equ. 245, 1883–1922 (2008)
Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton (2006)
Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)
Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Volume I: Theory. Kluwer Academic Publishers, Dordrecht (1997)
Kyritsi, S., Papageorgiou, N.S.: Multiple solutions for superlinear Dirichlet problems with an indefinite potential. Ann. Mat. Pura Appl. 192, 297–315 (2013)
Li, C., Li, S., Liu, J.: Splitting theorem, Poincaré–Hopf theorem and jumping nonlinear problems. J. Funct. Anal. 221, 439–455 (2005)
Michael, E.: Continuous selections I. Ann. Math. 63, 361–382 (1956)
Moroz, V.: Solutions of superlinear at zero elliptic equations via Morse theory. Topol. Methods Nonlinear Anal. 10, 1–11 (1997)
Motreanu, D., Motreanu, V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)
Mugnai, D., Papageorgiou, N.S.: Resonant nonlinear Neumann problems with indefinite weight. Ann. Sc. Norm. Super. Pisa 11, 729–788 (2012)
Palais, R.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966)
Papageorgiou, N.S., Kyritsi, S.: Handbook of Applied Analysis. Springer, New York (2009)
Papageorgiou, N.S., Papalini, F.: Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries. Isr. J. Math. 201, 761–796 (2014)
Papageorgiou, N.S., Rădulescu, V.D.: Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity. Contemp. Math. 595, 293–315 (2013)
Papageorgiou, N.S., Rădulescu, V.D.: Multiple solutions with precise sign information for parametric Robin problems. J. Differ. Equ. 256, 2449–2479 (2014)
Papageorgiou, N.S., Rădulescu, V.D.: Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential. Trans. Am. Math. Soc. doi:10.1090/S0002-9947-2014-06518-5
Papageorgiou, N.S., Smyrlis, G.: On a class of parametric Neumann problems with indefinite and unbounded potetial. Forum Math. doi:10.1515/forum-2012-0042
Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)
Struwe, M.: A note on a result of Ambrosetti and Mancini. Ann. Mat. Pura Appl. 81, 107–115 (1982)
Struwe, M.: Variational Methods. Springer, Berlin (1990)
Wang, X.: Neumann problems of semilinear elliptic equations involving critical Sobolev exponents. J. Differ. Equ. 93, 283–310 (1991)
Acknowledgments
The authors wish to thank two very knowledgeable referees for their corrections and helpful remarks which improved the paper considerably. V. Rǎdulescu acknowledges the support through Grant of the Executive Council for Funding Higher Education, Research and Innovation, Romania-UEFISCDI, Project Type: Advanced Collaborative Research Projects - PCCA, No 23/2014.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Papageorgiou, N.S., Rădulescu, V.D. Robin problems with indefinite, unbounded potential and reaction of arbitrary growth. Rev Mat Complut 29, 91–126 (2016). https://doi.org/10.1007/s13163-015-0181-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13163-015-0181-y
Keywords
- Indefinite and unbounded potential
- Robin boundary condition
- Constant sign and nodal solutions
- Multiplicity theorem
- Critical groups