Boundary concentrations on segments for a Neumann Ambrosetti-Prodi problem

Weiwei Ao; Mengdie Fu; Chao Liu

doi:10.3934/dcds.2022083

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

Boundary concentrations on segments for a Neumann Ambrosetti-Prodi problem

Weiwei Ao ^1, , Mengdie Fu ^1, and Chao Liu ^2,

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].

Keywords: Ambrosetti-Prodi equation, boundary concentrating solutions, Lyapunov-Schmidt reduction.

Mathematics Subject Classification: Primary: 35J10, 35J61.

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.

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Boundary concentrations on segments for a Neumann Ambrosetti-Prodi problem

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