July  2016, 15(4): 1107-1123. doi: 10.3934/cpaa.2016.15.1107

Nonlinear noncoercive Neumann problems

1. 

Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków

2. 

Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków, Poland

3. 

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

Received  August 2014 Revised  February 2016 Published  April 2016

We consider nonlinear, nonhomogeneous and noncoercive Neumann problems with a Carathéodory reaction which is either $(p-1)$-superlinear near $\pm\infty$ (without satisfying the usual in such cases Ambrosetti-Rabinowitz condition) or $(p-1)$-sublinear near $\pm\infty$. Using variational methods and Morse theory (critical groups) we prove two existence theorems.
Citation: Leszek Gasiński, Liliana Klimczak, Nikolaos S. Papageorgiou. Nonlinear noncoercive Neumann problems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1107-1123. doi: 10.3934/cpaa.2016.15.1107
References:
[1]

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349.

[2]

T. Bartsch and S.-J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance,, \emph{Nonlinear Anal.}, 28 (1997), 419. doi: 10.1016/0362-546X(95)00167-T.

[3]

J. Dugundji, Topology,, Allyn and Bacon, (1978).

[4]

M. Filippakis and N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the $p$-Laplacian,, \emph{J. Differential Equations}, 245 (2008), 1883. doi: 10.1016/j.jde.2008.07.004.

[5]

L. Gasiński and N.S. Papageorgiou, Nonlinear Analysis,, Chapman and Hall/CRC Press, (2006).

[6]

L. Gasiński and N.S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian-type equations,, \emph{Adv. Nonlinear Stud.}, 8 (2008), 843.

[7]

L. Gasiński and N.S. Papageorgiou, Multiple solutions for nonlinear Dirichlet problems with concave terms,, \emph{Math. Scand.}, 113 (2013), 206.

[8]

L. Gasiński and N.S. Papageorgiou, A pair of positive solutions for the Dirichlet $p(z)$-Laplacian with concave and convex nonlinearities,, \emph{J. Global Optim.}, 56 (2013), 1347. doi: 10.1007/s10898-011-9841-8.

[9]

L. Gasiński and N.S. Papageorgiou, On generalized logistic equations with a non-homogeneous differential operator,, \emph{Dyn. Syst.}, 29 (2014), 190. doi: 10.1080/14689367.2013.870125.

[10]

L. Gasiński and N.S. Papageorgiou, Positive solutions for parametric equidiffusive $p$-Laplacian equations,, \emph{Acta Math. Sci. Ser. B Engl. Ed.}, 34 (2014), 610. doi: 10.1016/S0252-9602(14)60033-3.

[11]

L. Gasiński and N.S. Papageorgiou, A pair of positive solutions for $(p,q)$-equations with combined nonlinearities,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 203. doi: 10.3934/cpaa.2014.13.203.

[12]

L. Gasiński and N.S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2037. doi: 10.3934/dcds.2014.34.2037.

[13]

L. Gasiński and N.S. Papageorgiou, Multiple solutions for a class of nonlinear Neumann eigenvalue problems,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 1491. doi: 10.3934/cpaa.2014.13.1491.

[14]

A. Granas and J. Dugundji, Fixed Point Theory,, Springer, (2003). doi: 10.1007/978-0-387-21593-8.

[15]

Q.-S. Jiu and J.-B. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian,, \emph{J. Math. Anal. Appl.}, 281 (2003), 587. doi: 10.1016/S0022-247X(03)00165-3.

[16]

G.M. Lieberman, The natural generalizations of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations,, \emph{Comm. Partial Differential Equations}, 16 (1991), 311. doi: 10.1080/03605309108820761.

[17]

V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory,, \emph{Topol. Methods Nonlinear Anal.}, 10 (1997), 387.

[18]

D. Motreanu, V.V. Motreanu and N.S. Papageorgiou, A unified approach for multiple constant sign and nodal solutions,, \emph{Adv. Differential Equations}, 121 (2007), 1363.

[19]

D. Motreanu, V. Motreanu and N.S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems,, Springer, (2014). doi: 10.1007/978-1-4614-9323-5.

[20]

Z.-Q. Wang, On a superlinear elliptic equation,, \emph{Ann. Inst. H. Poincar{\'e} Anal. Non Lin{\'e}aire}, 8 (1991), 43.

show all references

References:
[1]

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349.

[2]

T. Bartsch and S.-J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance,, \emph{Nonlinear Anal.}, 28 (1997), 419. doi: 10.1016/0362-546X(95)00167-T.

[3]

J. Dugundji, Topology,, Allyn and Bacon, (1978).

[4]

M. Filippakis and N.S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the $p$-Laplacian,, \emph{J. Differential Equations}, 245 (2008), 1883. doi: 10.1016/j.jde.2008.07.004.

[5]

L. Gasiński and N.S. Papageorgiou, Nonlinear Analysis,, Chapman and Hall/CRC Press, (2006).

[6]

L. Gasiński and N.S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian-type equations,, \emph{Adv. Nonlinear Stud.}, 8 (2008), 843.

[7]

L. Gasiński and N.S. Papageorgiou, Multiple solutions for nonlinear Dirichlet problems with concave terms,, \emph{Math. Scand.}, 113 (2013), 206.

[8]

L. Gasiński and N.S. Papageorgiou, A pair of positive solutions for the Dirichlet $p(z)$-Laplacian with concave and convex nonlinearities,, \emph{J. Global Optim.}, 56 (2013), 1347. doi: 10.1007/s10898-011-9841-8.

[9]

L. Gasiński and N.S. Papageorgiou, On generalized logistic equations with a non-homogeneous differential operator,, \emph{Dyn. Syst.}, 29 (2014), 190. doi: 10.1080/14689367.2013.870125.

[10]

L. Gasiński and N.S. Papageorgiou, Positive solutions for parametric equidiffusive $p$-Laplacian equations,, \emph{Acta Math. Sci. Ser. B Engl. Ed.}, 34 (2014), 610. doi: 10.1016/S0252-9602(14)60033-3.

[11]

L. Gasiński and N.S. Papageorgiou, A pair of positive solutions for $(p,q)$-equations with combined nonlinearities,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 203. doi: 10.3934/cpaa.2014.13.203.

[12]

L. Gasiński and N.S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2037. doi: 10.3934/dcds.2014.34.2037.

[13]

L. Gasiński and N.S. Papageorgiou, Multiple solutions for a class of nonlinear Neumann eigenvalue problems,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 1491. doi: 10.3934/cpaa.2014.13.1491.

[14]

A. Granas and J. Dugundji, Fixed Point Theory,, Springer, (2003). doi: 10.1007/978-0-387-21593-8.

[15]

Q.-S. Jiu and J.-B. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian,, \emph{J. Math. Anal. Appl.}, 281 (2003), 587. doi: 10.1016/S0022-247X(03)00165-3.

[16]

G.M. Lieberman, The natural generalizations of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations,, \emph{Comm. Partial Differential Equations}, 16 (1991), 311. doi: 10.1080/03605309108820761.

[17]

V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory,, \emph{Topol. Methods Nonlinear Anal.}, 10 (1997), 387.

[18]

D. Motreanu, V.V. Motreanu and N.S. Papageorgiou, A unified approach for multiple constant sign and nodal solutions,, \emph{Adv. Differential Equations}, 121 (2007), 1363.

[19]

D. Motreanu, V. Motreanu and N.S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems,, Springer, (2014). doi: 10.1007/978-1-4614-9323-5.

[20]

Z.-Q. Wang, On a superlinear elliptic equation,, \emph{Ann. Inst. H. Poincar{\'e} Anal. Non Lin{\'e}aire}, 8 (1991), 43.

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