# American Institute of Mathematical Sciences

October  2022, 42(10): 4991-5015. doi: 10.3934/dcds.2022083

## Boundary concentrations on segments for a Neumann Ambrosetti-Prodi problem

 1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 2 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510405, China

Received  January 2022 Revised  March 2022 Published  October 2022 Early access  June 2022

Fund Project: The research of the first author is supported by NSFC 12071357 and 12131017

Given a smooth bounded domain
 $\Omega\subset{{\mathbb R}}^2$
, we consider the following Ambrosetti-Prodi problem with Neumann boundary:
 $\begin{equation*} \left\{\begin{array}{l} -\Delta u = \left\vert{u}\right\vert^p-\sigma \quad {\mbox {in}} \ \Omega,\\ {\partial u \over \partial \nu} = 0 \quad {\mbox {on}} \ \partial \Omega. \end{array} \right. \end{equation*}$
where
 $p>2$
,
 $\sigma>0$
is a large parameter and
 $\nu$
denotes the outward normal of
 $\partial \Omega$
. We constructed a new class of solutions comprised of a large number of spikes concentrated on a segment of the boundary containing a local minimum point of the mean curvature function and having the same mean curvature at the endpoints. A similar boundary-concentrating phenomenon was obtained for the Lin-Ni-Takagi problem by Ao et al. [3].
Citation: Weiwei Ao, Mengdie Fu, Chao Liu. Boundary concentrations on segments for a Neumann Ambrosetti-Prodi problem. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4991-5015. doi: 10.3934/dcds.2022083
##### References:
 [1] H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburg Sect.A, 84 (1979), 145-151.  doi: 10.1017/S0308210500017017. [2] A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl., 93 (1972), 231-246.  doi: 10.1007/BF02412022. [3] W. Ao, H. Chan and J. Wei, Boundary concentrations on segments for the Lin-Ni-Takagi problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 653-696. [4] W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem, J. Differential Equations, 251 (2011), 881-901. [5] W. Ao, M. Musso and J. Wei, Triple junction solutions for a singularly perturbed Neumann problem, SIAM J. Math. Anal., 43 (2011), 2519-2541. [6] W. Ao, J. Wei and J. Zeng, An optimal bound on the number of interior spike solutions for the Lin-Ni-Takagi problem, J. Funct. Anal., 265 (2013), 1324-1356.  doi: 10.1016/j.jfa.2013.06.016. [7] I. Bendahou, Z. Khemiri and F. Mahmoudi, On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem, Discrete Contin. Dyn. Syst., 40 (2020), 2367-2391.  doi: 10.3934/dcds.2020118. [8] H. Berestycki, Le nombre de solutions de certains problemes semi lineaires elliptiques, J. Func. Anal., 40 (1981), 1-29.  doi: 10.1016/0022-1236(81)90069-0. [9] H. Berestycki and P. L. Lions, Sharp existence results for a class of semilinear elliptic problems, Bol. Soc. Bras. Mat., 12 (1981), 9-19. [10] M. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J., 24 (1974), 837-846.  doi: 10.1512/iumj.1975.24.24066. [11] A. Butscher and R. Mazzeo, CMC hypersurfaces condensing to geodesic segments and rays in Riemannian manifolds, Ann. Sc. Norm. Super. Pisa. Cl. Sci. (5), 11 (2012), 653-706. [12] E. N. Dancer, On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math., 25 (1995), 957-975.  doi: 10.1216/rmjm/1181072198. [13] E. N. Dancer, A note on asymptotic uniqueness for some non linearities which change sign, Bull.Aust. Math. Soc., 61 (2000), 305-312.  doi: 10.1017/S0004972700022309. [14] E. N. Dancer, On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pure Appl., 57 (1978), 351-366. [15] E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: The nonhomogeneous case, Adv. Differential Equations, 12 (2007), 961-993. [16] E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differential Equations, 210 (2005), 317-351.  doi: 10.1016/j.jde.2004.07.017. [17] E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture. Ⅱ, Comm. in Partial Differential Equations, 30 (2005), 1331-1358.  doi: 10.1080/03605300500258865. [18] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry or positive solutions or nonlinear elliptic equations in ${{\mathbb R}}^n$, Mathematical Analysis and Applications. Part A, Adv. Math. Suppl, (1981). [19] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin), 12 (1972), 30-39.  doi: 10.1007/BF00289234. [20] C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Can. J. Math., 52 (2000), 522-538.  doi: 10.4153/CJM-2000-024-x. [21] C. Gui and J. Wei, Multiple interior spike solutions for some singular perturbed Neumann problems, J. Differenrial Equations., 158 (1999), 1-27.  doi: 10.1016/S0022-0396(99)80016-3. [22] C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincare Anal. Non Lineaire, 17 (2000), 47-82.  doi: 10.1016/s0294-1449(99)00104-3. [23] P. Hess and B. Ruf, On a superlinear elliptic boundary value problem, Math. Z., 164 (1978), 9-14.  doi: 10.1007/BF01214785. [24] L. Hollman and P. J. McKenna, A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: Some numerical evidence, Commu. Pure Appl. Annl., 10 (2011), 785-802. [25] 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. [26] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [27] Z. Khemiri, F. Mahmoudi and A. Messaoudi, Concentration on submanifolds for an Ambrosetti-Prodi type problem, Calc. Var. Partial Differential Equations, 56 (2017), Paper No.19, 40 pp. doi: 10.1007/s00526-017-1117-9. [28] M. Kwong and L. Zhang, Uniqueness of positive solutions of $\Delta u+f(u) = 0$ in an annulus, Differential Integral Equations, 4 (1991), 583-599. [29] G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth. Ⅱ, J. Differential Equations, 227 (2006), 301-332.  doi: 10.1016/j.jde.2006.02.011. [30] F.-H. Lin, W.-M. Ni and J.-C. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.  doi: 10.1002/cpa.20139. [31] A. Malchiod and M. Montenegro, Boundary concentration phenomena for a singularly perturbed ellptic problem, Comm. Pure Appl. Math., 55 (2002), 1507-1508.  doi: 10.1002/cpa.10049. [32] A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.  doi: 10.1215/S0012-7094-04-12414-5. [33] A. Malchiodi and M. Montenegro, Boundary layers of arbitrary dimension for a singularly perturbed Neumann problem, Mat. Contemp., 27 (2004), 117-146. [34] B. B. Manna and S. Santra, On the Hollman McKenna conjecture: Interior concentration near curves, Discrete Contin. Dyn. Syst., 36 (2016), 5595-5626.  doi: 10.3934/dcds.2016046. [35] 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. [36] J. Mawhin, Ambrosetti-Prodi type results in nonlinear boundary value problems, Differential Equations and Mathematical Physics, Lecture Notes in Math., 1285 (1987), 290-313.  doi: 10.1007/BFb0080609. [37] J. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Differential Equations, 134 (1997), 104-133.  doi: 10.1006/jdeq.1996.3218. [38] J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492.  doi: 10.1016/s0294-1449(98)80031-0. [39] J. Wei and J. Yang, Concentration on lines for a singularly perturbed Neumann problem in two-dimensional domains, Indiana Univ. Math. J., 56 (2007), 3025-3073.  doi: 10.1512/iumj.2007.56.3133. [40] J. Wei and J. Yang, Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain, Discrete Contin. Dyn. Syst., 22 (2008), 465-508.  doi: 10.3934/dcds.2008.22.465.

show all references

##### References:
 [1] H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburg Sect.A, 84 (1979), 145-151.  doi: 10.1017/S0308210500017017. [2] A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl., 93 (1972), 231-246.  doi: 10.1007/BF02412022. [3] W. Ao, H. Chan and J. Wei, Boundary concentrations on segments for the Lin-Ni-Takagi problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 653-696. [4] W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem, J. Differential Equations, 251 (2011), 881-901. [5] W. Ao, M. Musso and J. Wei, Triple junction solutions for a singularly perturbed Neumann problem, SIAM J. Math. Anal., 43 (2011), 2519-2541. [6] W. Ao, J. Wei and J. Zeng, An optimal bound on the number of interior spike solutions for the Lin-Ni-Takagi problem, J. Funct. Anal., 265 (2013), 1324-1356.  doi: 10.1016/j.jfa.2013.06.016. [7] I. Bendahou, Z. Khemiri and F. Mahmoudi, On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem, Discrete Contin. Dyn. Syst., 40 (2020), 2367-2391.  doi: 10.3934/dcds.2020118. [8] H. Berestycki, Le nombre de solutions de certains problemes semi lineaires elliptiques, J. Func. Anal., 40 (1981), 1-29.  doi: 10.1016/0022-1236(81)90069-0. [9] H. Berestycki and P. L. Lions, Sharp existence results for a class of semilinear elliptic problems, Bol. Soc. Bras. Mat., 12 (1981), 9-19. [10] M. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J., 24 (1974), 837-846.  doi: 10.1512/iumj.1975.24.24066. [11] A. Butscher and R. Mazzeo, CMC hypersurfaces condensing to geodesic segments and rays in Riemannian manifolds, Ann. Sc. Norm. Super. Pisa. Cl. Sci. (5), 11 (2012), 653-706. [12] E. N. Dancer, On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math., 25 (1995), 957-975.  doi: 10.1216/rmjm/1181072198. [13] E. N. Dancer, A note on asymptotic uniqueness for some non linearities which change sign, Bull.Aust. Math. Soc., 61 (2000), 305-312.  doi: 10.1017/S0004972700022309. [14] E. N. Dancer, On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pure Appl., 57 (1978), 351-366. [15] E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: The nonhomogeneous case, Adv. Differential Equations, 12 (2007), 961-993. [16] E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differential Equations, 210 (2005), 317-351.  doi: 10.1016/j.jde.2004.07.017. [17] E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture. Ⅱ, Comm. in Partial Differential Equations, 30 (2005), 1331-1358.  doi: 10.1080/03605300500258865. [18] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry or positive solutions or nonlinear elliptic equations in ${{\mathbb R}}^n$, Mathematical Analysis and Applications. Part A, Adv. Math. Suppl, (1981). [19] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin), 12 (1972), 30-39.  doi: 10.1007/BF00289234. [20] C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Can. J. Math., 52 (2000), 522-538.  doi: 10.4153/CJM-2000-024-x. [21] C. Gui and J. Wei, Multiple interior spike solutions for some singular perturbed Neumann problems, J. Differenrial Equations., 158 (1999), 1-27.  doi: 10.1016/S0022-0396(99)80016-3. [22] C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincare Anal. Non Lineaire, 17 (2000), 47-82.  doi: 10.1016/s0294-1449(99)00104-3. [23] P. Hess and B. Ruf, On a superlinear elliptic boundary value problem, Math. Z., 164 (1978), 9-14.  doi: 10.1007/BF01214785. [24] L. Hollman and P. J. McKenna, A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: Some numerical evidence, Commu. Pure Appl. Annl., 10 (2011), 785-802. [25] 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. [26] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [27] Z. Khemiri, F. Mahmoudi and A. Messaoudi, Concentration on submanifolds for an Ambrosetti-Prodi type problem, Calc. Var. Partial Differential Equations, 56 (2017), Paper No.19, 40 pp. doi: 10.1007/s00526-017-1117-9. [28] M. Kwong and L. Zhang, Uniqueness of positive solutions of $\Delta u+f(u) = 0$ in an annulus, Differential Integral Equations, 4 (1991), 583-599. [29] G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth. Ⅱ, J. Differential Equations, 227 (2006), 301-332.  doi: 10.1016/j.jde.2006.02.011. [30] F.-H. Lin, W.-M. Ni and J.-C. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.  doi: 10.1002/cpa.20139. [31] A. Malchiod and M. Montenegro, Boundary concentration phenomena for a singularly perturbed ellptic problem, Comm. Pure Appl. Math., 55 (2002), 1507-1508.  doi: 10.1002/cpa.10049. [32] A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.  doi: 10.1215/S0012-7094-04-12414-5. [33] A. Malchiodi and M. Montenegro, Boundary layers of arbitrary dimension for a singularly perturbed Neumann problem, Mat. Contemp., 27 (2004), 117-146. [34] B. B. Manna and S. Santra, On the Hollman McKenna conjecture: Interior concentration near curves, Discrete Contin. Dyn. Syst., 36 (2016), 5595-5626.  doi: 10.3934/dcds.2016046. [35] 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. [36] J. Mawhin, Ambrosetti-Prodi type results in nonlinear boundary value problems, Differential Equations and Mathematical Physics, Lecture Notes in Math., 1285 (1987), 290-313.  doi: 10.1007/BFb0080609. [37] J. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Differential Equations, 134 (1997), 104-133.  doi: 10.1006/jdeq.1996.3218. [38] J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492.  doi: 10.1016/s0294-1449(98)80031-0. [39] J. Wei and J. Yang, Concentration on lines for a singularly perturbed Neumann problem in two-dimensional domains, Indiana Univ. Math. J., 56 (2007), 3025-3073.  doi: 10.1512/iumj.2007.56.3133. [40] J. Wei and J. Yang, Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain, Discrete Contin. Dyn. Syst., 22 (2008), 465-508.  doi: 10.3934/dcds.2008.22.465.
 [1] Imene Bendahou, Zied Khemiri, Fethi Mahmoudi. On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2367-2391. doi: 10.3934/dcds.2020118 [2] Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201 [3] Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739 [4] F. R. Pereira. Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents. Communications on Pure and Applied Analysis, 2008, 7 (2) : 355-372. doi: 10.3934/cpaa.2008.7.355 [5] 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 [6] Vincenzo Ambrosio. Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 781-815. doi: 10.3934/dcds.2020062 [7] Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4487-4518. doi: 10.3934/dcds.2019184 [8] Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. PDE problems with concentrating terms near the boundary. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2147-2195. doi: 10.3934/cpaa.2020095 [9] Ezzeddine Zahrouni. On the Lyapunov functions for the solutions of the generalized Burgers equation. Communications on Pure and Applied Analysis, 2003, 2 (3) : 391-410. doi: 10.3934/cpaa.2003.2.391 [10] Liping Wang, Dong Ye. Concentrating solutions for an anisotropic elliptic problem with large exponent. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3771-3797. doi: 10.3934/dcds.2015.35.3771 [11] Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5125-5147. doi: 10.3934/dcds.2019208 [12] José M. Arrieta, Ariadne Nogueira, Marcone C. Pereira. Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4217-4246. doi: 10.3934/dcdsb.2019079 [13] Vincenzo Ambrosio. Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2265-2284. doi: 10.3934/dcds.2017099 [14] Long Wei. Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions. Communications on Pure and Applied Analysis, 2008, 7 (4) : 925-946. doi: 10.3934/cpaa.2008.7.925 [15] Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control and Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015 [16] Dimitri Breda, Sara Della Schiava. Pseudospectral reduction to compute Lyapunov exponents of delay differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2727-2741. doi: 10.3934/dcdsb.2018092 [17] Eun Bee Choi, Yun-Ho Kim. Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition. Conference Publications, 2015, 2015 (special) : 276-286. doi: 10.3934/proc.2015.0276 [18] Yang Wang. The maximal number of interior peak solutions concentrating on hyperplanes for a singularly perturbed Neumann problem. Communications on Pure and Applied Analysis, 2011, 10 (2) : 731-744. doi: 10.3934/cpaa.2011.10.731 [19] Yi He, Gongbao Li. Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 731-762. doi: 10.3934/dcds.2016.36.731 [20] M. Sumon Hossain, M. Monir Uddin. Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations. Numerical Algebra, Control and Optimization, 2019, 9 (2) : 173-186. doi: 10.3934/naco.2019013

2021 Impact Factor: 1.588

## Metrics

• PDF downloads (68)
• HTML views (32)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]