# American Institute of Mathematical Sciences

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.

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##### 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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