April  2018, 11(2): 345-355. doi: 10.3934/dcdss.2018019

Ambrosetti-Prodi type result to a Neumann problem via a topological approach

Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206,33100 Udine, Italy

Received  February 2017 Revised  May 2017 Published  January 2018

Fund Project: Work partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Progetto di Ricerca 2016: "Problemi differenziali non lineari: esistenza, molteplicità e proprietà qualitative delle soluzioni".

We prove an Ambrosetti-Prodi type result for a Neumann problem associated to the equation $u''+f(x, u(x))=μ$ when the nonlinearity has the following form:$f(x, u):=a(x)g(u)-p(x)$. The assumptions considered generalize the classical one, $f(x, u)\to+∞$ as $|u|\to+∞$, without requiring any uniformity condition in $x$. The multiplicity result which characterizes these kind of problems will be proved by means of the shooting method.

Citation: Elisa Sovrano. Ambrosetti-Prodi type result to a Neumann problem via a topological approach. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 345-355. doi: 10.3934/dcdss.2018019
References:
[1]

H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 145-151.  doi: 10.1017/S0308210500017017.

[2]

A. Ambrosetti, Observations on global inversion theorems, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 22 (2011), 3-15.  doi: 10.4171/RLM/584.

[3]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. (4), 93 (1972), 231-246.  doi: 10.1007/BF02412022.

[4]

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis vol. 34 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1993.

[5]

H. Berestycki and P. L. Lions, Sharp existence results for a class of semilinear elliptic problems, Bol. Soc. Brasil. Mat., 12 (1981), 9-19.  doi: 10.1007/BF02588317.

[6]

M. S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J., 24 (1974/75), 837-846.  doi: 10.1512/iumj.1975.24.24066.

[7]

E. N. Dancer, On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pures Appl. (9), 57 (1978), 351-366. 

[8]

C. De Coster and P. Habets, Two-point Boundary Value Problems: Lower and Upper Solutions vol. 205 of Mathematics in Science and Engineering, Elsevier B. V. , Amsterdam, 2006.

[9]

D. G. de Figueiredo, Lectures on Boundary Value Problems of Ambrosetti-Prodi Type Atas do 12o Seminario Brasileiro de Análise, São Paulo, 1980.

[10]

C. FabryJ. Mawhin and M. N. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations, Bull. London Math. Soc., 18 (1986), 173-180.  doi: 10.1112/blms/18.2.173.

[11]

A. Fonda and A. Sfecci, On a singular periodic Ambrosetti-Prodi problem, Nonlinear Anal., 149 (2017), 146-155.  doi: 10.1016/j.na.2016.10.018.

[12]

S. Fučík, Boundary value problems with jumping nonlinearities, Časopis Pěst. Mat., 101 (1976), 69-87. 

[13]

J. L. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), 567-597.  doi: 10.1002/cpa.3160280502.

[14]

A. Manes and A. M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), 7 (1973), 285-301. 

[15]

J. Mawhin, Ambrosetti-Prodi type results in nonlinear boundary value problems, in Differential equations and mathematical physics (Birmingham, Ala. , 1986), vol. 1285 of Lecture Notes in Math. , Springer, Berlin, 1987,290-313. doi: 10.1007/BFb0080609.

[16]

J. Mawhin, The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the p-Laplacian, J. Eur. Math. Soc. (JEMS), 8 (2006), 375-388.  doi: 10.4171/JEMS/58.

[17]

R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Ital. B (7), 3 (1989), 533-546. 

[18]

R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type, Differential Integral Equations, 3 (1990), 275-284. 

[19]

A. E. Presoto and F. O. de Paiva, A Neumann problem of Ambrosetti-Prodi type, J. Fixed Point Theory Appl., 18 (2016), 189-200.  doi: 10.1007/s11784-015-0277-5.

[20]

I. Rachůnková, On the number of solutions of the Neumann problem for the ordinary second order differential equation, Ann. Math. Sil., 7 (1993), 79-87. 

[21]

E. Sovrano and F. Zanolin, The Ambrosetti-Prodi periodic problem: Different routes to complex dynamics, Dynam. Systems Appl. (to appear).

show all references

References:
[1]

H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 145-151.  doi: 10.1017/S0308210500017017.

[2]

A. Ambrosetti, Observations on global inversion theorems, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 22 (2011), 3-15.  doi: 10.4171/RLM/584.

[3]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. (4), 93 (1972), 231-246.  doi: 10.1007/BF02412022.

[4]

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis vol. 34 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1993.

[5]

H. Berestycki and P. L. Lions, Sharp existence results for a class of semilinear elliptic problems, Bol. Soc. Brasil. Mat., 12 (1981), 9-19.  doi: 10.1007/BF02588317.

[6]

M. S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J., 24 (1974/75), 837-846.  doi: 10.1512/iumj.1975.24.24066.

[7]

E. N. Dancer, On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pures Appl. (9), 57 (1978), 351-366. 

[8]

C. De Coster and P. Habets, Two-point Boundary Value Problems: Lower and Upper Solutions vol. 205 of Mathematics in Science and Engineering, Elsevier B. V. , Amsterdam, 2006.

[9]

D. G. de Figueiredo, Lectures on Boundary Value Problems of Ambrosetti-Prodi Type Atas do 12o Seminario Brasileiro de Análise, São Paulo, 1980.

[10]

C. FabryJ. Mawhin and M. N. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations, Bull. London Math. Soc., 18 (1986), 173-180.  doi: 10.1112/blms/18.2.173.

[11]

A. Fonda and A. Sfecci, On a singular periodic Ambrosetti-Prodi problem, Nonlinear Anal., 149 (2017), 146-155.  doi: 10.1016/j.na.2016.10.018.

[12]

S. Fučík, Boundary value problems with jumping nonlinearities, Časopis Pěst. Mat., 101 (1976), 69-87. 

[13]

J. L. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), 567-597.  doi: 10.1002/cpa.3160280502.

[14]

A. Manes and A. M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), 7 (1973), 285-301. 

[15]

J. Mawhin, Ambrosetti-Prodi type results in nonlinear boundary value problems, in Differential equations and mathematical physics (Birmingham, Ala. , 1986), vol. 1285 of Lecture Notes in Math. , Springer, Berlin, 1987,290-313. doi: 10.1007/BFb0080609.

[16]

J. Mawhin, The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the p-Laplacian, J. Eur. Math. Soc. (JEMS), 8 (2006), 375-388.  doi: 10.4171/JEMS/58.

[17]

R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Ital. B (7), 3 (1989), 533-546. 

[18]

R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type, Differential Integral Equations, 3 (1990), 275-284. 

[19]

A. E. Presoto and F. O. de Paiva, A Neumann problem of Ambrosetti-Prodi type, J. Fixed Point Theory Appl., 18 (2016), 189-200.  doi: 10.1007/s11784-015-0277-5.

[20]

I. Rachůnková, On the number of solutions of the Neumann problem for the ordinary second order differential equation, Ann. Math. Sil., 7 (1993), 79-87. 

[21]

E. Sovrano and F. Zanolin, The Ambrosetti-Prodi periodic problem: Different routes to complex dynamics, Dynam. Systems Appl. (to appear).

Figure 1.  Numerical simulations for the Neumann problem $(\mathcal{P}_{\mu})$ defined as in Example
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