# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - S, 2018, 11 (2) : 345-355. doi: 10.3934/dcdss.2018019
##### References:

show all references

##### References:
Numerical simulations for the Neumann problem $(\mathcal{P}_{\mu})$ defined as in Example
 [1] F. R. Pereira. Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents. Communications on Pure & Applied Analysis, 2008, 7 (2) : 355-372. doi: 10.3934/cpaa.2008.7.355 [2] Antonio Iannizzotto, Nikolaos S. Papageorgiou. Existence and multiplicity results for resonant fractional boundary value problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 511-532. doi: 10.3934/dcdss.2018028 [3] Patrick Winkert. Multiplicity results for a class of elliptic problems with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (2) : 785-802. doi: 10.3934/cpaa.2013.12.785 [4] Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061 [5] Giuseppina Barletta, Gabriele Bonanno. Multiplicity results to elliptic problems in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 715-727. doi: 10.3934/dcdss.2012.5.715 [6] Eunkyoung Ko, Eun Kyoung Lee, R. Shivaji. Multiplicity results for classes of singular problems on an exterior domain. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5153-5166. doi: 10.3934/dcds.2013.33.5153 [7] Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4487-4518. doi: 10.3934/dcds.2019184 [8] Monica Lazzo. Existence and multiplicity results for a class of nonlinear elliptic problems in $\mathbb(R)^N$. Conference Publications, 2003, 2003 (Special) : 526-535. doi: 10.3934/proc.2003.2003.526 [9] Yosra Boukari, Houssem Haddar. The factorization method applied to cracks with impedance boundary conditions. Inverse Problems & Imaging, 2013, 7 (4) : 1123-1138. doi: 10.3934/ipi.2013.7.1123 [10] B. Abdellaoui, E. Colorado, I. Peral. Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions. Communications on Pure & Applied Analysis, 2006, 5 (1) : 29-54. doi: 10.3934/cpaa.2006.5.29 [11] Alexandre Nolasco de Carvalho, Marcos Roberto Teixeira Primo. Spatial homogeneity in parabolic problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2004, 3 (4) : 637-651. doi: 10.3934/cpaa.2004.3.637 [12] Giuseppe Maria Coclite, Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Continuous dependence in hyperbolic problems with Wentzell boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (1) : 419-433. doi: 10.3934/cpaa.2014.13.419 [13] Marek Galewski, Shapour Heidarkhani, Amjad Salari. Multiplicity results for discrete anisotropic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 203-218. doi: 10.3934/dcdsb.2018014 [14] J. Giacomoni, K. Sreeandh. Multiplicity results for a singular and quazilinear equation. Conference Publications, 2007, 2007 (Special) : 429-435. doi: 10.3934/proc.2007.2007.429 [15] Davide Guidetti. Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1401-1417. doi: 10.3934/cpaa.2016.15.1401 [16] Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729 [17] Frederic Rousset. The residual boundary conditions coming from the real vanishing viscosity method. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 605-625. doi: 10.3934/dcds.2002.8.606 [18] Julián López-Gómez, Paul H. Rabinowitz. The effects of spatial heterogeneities on some multiplicity results. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 941-952. doi: 10.3934/dcds.2016.36.941 [19] Fábio R. Pereira. Multiplicity results for fractional systems crossing high eigenvalues. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2069-2088. doi: 10.3934/cpaa.2017102 [20] Alejandro Allendes, Alexander Quaas. Multiplicity results for extremal operators through bifurcation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 51-65. doi: 10.3934/dcds.2011.29.51

2018 Impact Factor: 0.545