On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem

Imene Bendahou; Zied Khemiri; Fethi Mahmoudi

doi:10.3934/dcds.2020118

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April 2020, 40(4): 2367-2391. doi: 10.3934/dcds.2020118

On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem

Imene Bendahou , Zied Khemiri and Fethi Mahmoudi ^,

Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Campus Universitaire, 2092 Tunis El Manar, Tunisia

^* Corresponding author: F. Mahmoudi

Received July 2019 Revised November 2019 Published January 2020

Fund Project: F. Mahmoudi is supported by Fondecyt Grant 1180526, CONICYT.

Given a smooth bounded domain

$ \Omega \subset \mathbb {R}^n $

and consider the problem

$ \left\{\begin{array} {cccccc} - \Delta u = |u|^p - \sigma &\hbox{in } \Omega \\ \dfrac{\partial u}{\partial \nu} = 0 &\hbox{on}\ \partial \Omega \end{array}\right. $

where

$ p $

is subcritical exponent (

$ p > 1 $

$ n = 2 $

and

$ 1 < p < \frac{n+2}{n-2} $

$ n \geq 3 $

$ \sigma > 0 $

is a large parameter and

$ \nu $

denotes the outward normal of

$ \partial\Omega $

. Let

$ \Gamma $

be an interior straighline intersecting orthogonally with

$ \partial\Omega $

. Assuming moreover that

$ \Gamma $

satisfies a non-degeneracy condition, we construct a new class of solutions which consist of large number of spikes concentrating on

$ \Gamma $

, showing as in [5,6] that higher dimensional concentration can exist without resonance condition.

Keywords: Neumann Ambrosetti-Prodi type problem, concentration sets, the Lyapunov-Schmidt reduction method.

Mathematics Subject Classification: Primary: 35J20, 35J25; Secondary: 35J60, 35J61.

Citation: 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

References:

[1]	H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Ray. Soc. Edinburg Sect. A, 84 (1979), 145-151. doi: 10.1017/S0308210500017017.
[2]	A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. Ⅱ, Indiana Univ. Math. J., 53 (2004), 297-329. doi: 10.1512/iumj.2004.53.2400.
[3]	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.
[4]	A. Ambrosetti, E. Colorado and D. Ruiz, Multi-bump solutions to linearly coupled systems of nonlinear Schrödinger equations, Calculus of Variations. Calc. Var., 30 (2007), 85-112. doi: 10.1007/s00526-006-0079-0.
[5]	W. W. Ao, M. Musso and J. C. Wei, On spikes concentrating on line segments to a semilinear Neumann problem, Journal of Differential Equations, 251 (2011), 881-901. doi: 10.1016/j.jde.2011.05.009.
[6]	W. W. Ao, M. Musso and J. C. Wei, Triple junction solutions for a singularly perturbed Neumann problem, SIAM Journal on Mathematical Analysis, 43 (2011), 2519-2541. doi: 10.1137/100812100.
[7]	W. W. Ao, J. C. Wei and J. Zeng, An optimal bound on the number of interior spike solutions for the Lin-Ni-Takagi problem, Journal of Functional Analysis, 265 (2013), 1324-1356. doi: 10.1016/j.jfa.2013.06.016.
[8]	H. Berestycki, Le nombre de solutions de certains problémes semi linéaires 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. doi: 10.1007/BF02588317.
[10]	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.
[11]	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.
[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, On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pure Appl., 57 (1978), 351-366.
[14]	E. N. Dancer and S. S. Yan, Multipeak solutions for a singular perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262. doi: 10.2140/pjm.1999.189.241.
[15]	E. N. Dancer and S. S. Yan, On the superlinear Lazer-McKenna conjecture. Ⅱ, Comm. in Partial Differential Equations, 30 (2005), 1331-1358. doi: 10.1080/03605300500258865.
[16]	E. N. Dancer and S. 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. Santra, On the superlinear Lazer-McKenna conjecture: The nonhomogeneous case, Adv. Differential Equations, 12 (2007), 961-993.
[18]	M. del Pino, P. L. Felmer and J. C. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79. doi: 10.1137/S0036141098332834.
[19]	M. del Pino, P. L. Felmer and J. C. Wei, On the role of distance function in some singularly perturbed problems, Comm. PDE, 25 (2000), 155-177. doi: 10.1080/03605300008821511.
[20]	M. del Pino, P. L. Felmer and J. C. Wei, Mutiple peak solutions for some singular perturbation problems, Cal. Var. PDE, 10 (2000), 119-134. doi: 10.1007/s005260050147.
[21]	M. del Pino, F. Mahmoudi and M. Musso, Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents, Journal of the European Mathematical Society, 16 (2014), 1687-1748. doi: 10.4171/JEMS/473.
[22]	M. del Pino and M. Musso, Bubbling and criticality in two and higher dimensions, Recent Advances in Elliptic and Parabolic Problems, World Sci. Publ., Hackensack, NJ, (2005), 41–59. doi: 10.1142/9789812702050_0004.
[23]	M. del Pino, M. Musso and F. Pacard, Bubbling along geodesics near the second critical exponent, J. Eur. Math. Soc. (JEMS), 12 (2010), 1553-1605. doi: 10.4171/JEMS/241.
[24]	S. B. Deng, F. Mahmoudi and M. Musso, Bubbling on boundary sub-manifolds for a semilinear Neumann problem near high critical exponents, Discrete and Continuous Dynamical Systems, 36 (2016), 3035-3076. doi: 10.3934/dcds.2016.36.3035.
[25]	S. B. Deng, M. Musso and A. Pistoia, Concentration on minimal submanifolds for a Yamabe type problem, Communications in Partial Differential Equations, 41 (2016), 1379-1425. doi: 10.1080/03605302.2016.1209519.
[26]	A. Gierer and H.Meinhardt, A thoery of biological pattern formation, Kybernetik, 12 (1972), 30–39, http://dx.doi.org/10.1007/BF00289234.
[27]	M. Grossi, A. Pistoia and J. C. Wei, Existence of multi-peak solutions for a semi-linear Neumann problem via non-smooth critical point theory, Cal. Var. PDE, 11 (2000), 143-175. doi: 10.1007/PL00009907.
[28]	C.-F. Gui and J. C. 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.
[29]	C.-F. Gui, J. C. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 47-82. doi: 10.1016/S0294-1449(99)00104-3.
[30]	C.-F. Gui and J. C. Wei, Multiple interior spike solutions for some singular perturbed Neumann problems, J. Diff. Eqns., 158 (1999), 1-27. doi: 10.1016/S0022-0396(99)80016-3.
[31]	P. Hess and B. Ruf, On a superlinear elliptic boundary value problem, Math. Z., 164 (1978), 9-14. doi: 10.1007/BF01214785.
[32]	L. Hollman and P. J. McKenna, A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: Some numerical evidence, Commun. Pure Appl. Anal., 10 (2011), 785-802. doi: 10.3934/cpaa.2011.10.785.
[33]	N. Kapouleas, Complete CMC surfaces in Euclidean three-space, Ann. of Math., 131 (1990), 239-330. doi: 10.2307/1971494.
[34]	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.
[35]	Z. Khemiri, F. Mahmoudi and A. Messaoudi, Concentration on submanifolds for an Ambrosetti-Prodi type problem, Calc. Var. Partial Differential Equations, 56 (2017), Art. 19, 40 pp. doi: 10.1007/s00526-017-1117-9.
[36]	M. K. Kwong and L. Zhang, Uniqueness of positive solutions of $\Delta u + f(u) = 0$ in an annulus, Diferential Integral Equations, 4 (1991), 583-599.
[37]	A. C. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl., 84 (1981), 282-294. doi: 10.1016/0022-247X(81)90166-9.
[38]	G. B. Li, S. S. Yan and J. F. Yang, The super linear 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.
[39]	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.
[40]	C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.
[41]	F. Mahmoudi and A. Malchiodi, Concentration on minimal sub manifolds for a singularly perturbed Neumann problem, Adv. in Math., 209 (2007), 460-525. doi: 10.1016/j.aim.2006.05.014.
[42]	F. Mahmoudi and A. Malchiodi, Concentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 17 (2006), 279-290. doi: 10.4171/RLM/469.
[43]	A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, Geom. Funct. Anal., 15 (2005), 1162-1222. doi: 10.1007/s00039-005-0542-7.
[44]	A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\mathbb{R}^n$, Advances in Mathematics, 221 (2009), 1843-1909. doi: 10.1016/j.aim.2009.03.012.
[45]	A. Malchiodi 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.
[46]	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.
[47]	A. Malchiodi, W.-M. Ni and J. C. Wei, Multiple clustered layer solutions for semi-linear Neumann problems on a ball, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 143-163. doi: 10.1016/j.anihpc.2004.05.003.
[48]	B. B. Manna and S. Santra, On the Hollman McKenna conjecture: Interior concentration On Geodesics, Discrete Contin. Dyn. Syst., 36 (2016), 5595-5626.
[49]	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.
[50]	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.
[51]	R. Mazzeo and F. Pacard, CMC surfaces with Delaunay ends, Comm. Anal. Geom., 9 (2001), 169-237. doi: 10.4310/CAG.2001.v9.n1.a6.
[52]	W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices of Amer. Math. Soc., 45 (1998), 9-18.
[53]	W.-M. Ni and I. Takagi, On the shape of least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851. doi: 10.1002/cpa.3160440705.
[54]	W.-M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4.
[55]	W.-M. Ni and J. C. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768. doi: 10.1002/cpa.3160480704.
[56]	J. C. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Diff. Eqns., 134 (1997), 104-133. doi: 10.1006/jdeq.1996.3218.
[57]	J. C. Wei and S. S. Yan, Lazer-McKenna conjecture: The critical case, J. Funct. Anal., 244 (2007), 639-667. doi: 10.1016/j.jfa.2006.11.002.
[58]	J. C. 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.

show all references

References:

[1]	H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Ray. Soc. Edinburg Sect. A, 84 (1979), 145-151. doi: 10.1017/S0308210500017017.
[2]	A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. Ⅱ, Indiana Univ. Math. J., 53 (2004), 297-329. doi: 10.1512/iumj.2004.53.2400.
[3]	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.
[4]	A. Ambrosetti, E. Colorado and D. Ruiz, Multi-bump solutions to linearly coupled systems of nonlinear Schrödinger equations, Calculus of Variations. Calc. Var., 30 (2007), 85-112. doi: 10.1007/s00526-006-0079-0.
[5]	W. W. Ao, M. Musso and J. C. Wei, On spikes concentrating on line segments to a semilinear Neumann problem, Journal of Differential Equations, 251 (2011), 881-901. doi: 10.1016/j.jde.2011.05.009.
[6]	W. W. Ao, M. Musso and J. C. Wei, Triple junction solutions for a singularly perturbed Neumann problem, SIAM Journal on Mathematical Analysis, 43 (2011), 2519-2541. doi: 10.1137/100812100.
[7]	W. W. Ao, J. C. Wei and J. Zeng, An optimal bound on the number of interior spike solutions for the Lin-Ni-Takagi problem, Journal of Functional Analysis, 265 (2013), 1324-1356. doi: 10.1016/j.jfa.2013.06.016.
[8]	H. Berestycki, Le nombre de solutions de certains problémes semi linéaires 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. doi: 10.1007/BF02588317.
[10]	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.
[11]	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.
[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, On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pure Appl., 57 (1978), 351-366.
[14]	E. N. Dancer and S. S. Yan, Multipeak solutions for a singular perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262. doi: 10.2140/pjm.1999.189.241.
[15]	E. N. Dancer and S. S. Yan, On the superlinear Lazer-McKenna conjecture. Ⅱ, Comm. in Partial Differential Equations, 30 (2005), 1331-1358. doi: 10.1080/03605300500258865.
[16]	E. N. Dancer and S. 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. Santra, On the superlinear Lazer-McKenna conjecture: The nonhomogeneous case, Adv. Differential Equations, 12 (2007), 961-993.
[18]	M. del Pino, P. L. Felmer and J. C. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79. doi: 10.1137/S0036141098332834.
[19]	M. del Pino, P. L. Felmer and J. C. Wei, On the role of distance function in some singularly perturbed problems, Comm. PDE, 25 (2000), 155-177. doi: 10.1080/03605300008821511.
[20]	M. del Pino, P. L. Felmer and J. C. Wei, Mutiple peak solutions for some singular perturbation problems, Cal. Var. PDE, 10 (2000), 119-134. doi: 10.1007/s005260050147.
[21]	M. del Pino, F. Mahmoudi and M. Musso, Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents, Journal of the European Mathematical Society, 16 (2014), 1687-1748. doi: 10.4171/JEMS/473.
[22]	M. del Pino and M. Musso, Bubbling and criticality in two and higher dimensions, Recent Advances in Elliptic and Parabolic Problems, World Sci. Publ., Hackensack, NJ, (2005), 41–59. doi: 10.1142/9789812702050_0004.
[23]	M. del Pino, M. Musso and F. Pacard, Bubbling along geodesics near the second critical exponent, J. Eur. Math. Soc. (JEMS), 12 (2010), 1553-1605. doi: 10.4171/JEMS/241.
[24]	S. B. Deng, F. Mahmoudi and M. Musso, Bubbling on boundary sub-manifolds for a semilinear Neumann problem near high critical exponents, Discrete and Continuous Dynamical Systems, 36 (2016), 3035-3076. doi: 10.3934/dcds.2016.36.3035.
[25]	S. B. Deng, M. Musso and A. Pistoia, Concentration on minimal submanifolds for a Yamabe type problem, Communications in Partial Differential Equations, 41 (2016), 1379-1425. doi: 10.1080/03605302.2016.1209519.
[26]	A. Gierer and H.Meinhardt, A thoery of biological pattern formation, Kybernetik, 12 (1972), 30–39, http://dx.doi.org/10.1007/BF00289234.
[27]	M. Grossi, A. Pistoia and J. C. Wei, Existence of multi-peak solutions for a semi-linear Neumann problem via non-smooth critical point theory, Cal. Var. PDE, 11 (2000), 143-175. doi: 10.1007/PL00009907.
[28]	C.-F. Gui and J. C. 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.
[29]	C.-F. Gui, J. C. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 47-82. doi: 10.1016/S0294-1449(99)00104-3.
[30]	C.-F. Gui and J. C. Wei, Multiple interior spike solutions for some singular perturbed Neumann problems, J. Diff. Eqns., 158 (1999), 1-27. doi: 10.1016/S0022-0396(99)80016-3.
[31]	P. Hess and B. Ruf, On a superlinear elliptic boundary value problem, Math. Z., 164 (1978), 9-14. doi: 10.1007/BF01214785.
[32]	L. Hollman and P. J. McKenna, A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: Some numerical evidence, Commun. Pure Appl. Anal., 10 (2011), 785-802. doi: 10.3934/cpaa.2011.10.785.
[33]	N. Kapouleas, Complete CMC surfaces in Euclidean three-space, Ann. of Math., 131 (1990), 239-330. doi: 10.2307/1971494.
[34]	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.
[35]	Z. Khemiri, F. Mahmoudi and A. Messaoudi, Concentration on submanifolds for an Ambrosetti-Prodi type problem, Calc. Var. Partial Differential Equations, 56 (2017), Art. 19, 40 pp. doi: 10.1007/s00526-017-1117-9.
[36]	M. K. Kwong and L. Zhang, Uniqueness of positive solutions of $\Delta u + f(u) = 0$ in an annulus, Diferential Integral Equations, 4 (1991), 583-599.
[37]	A. C. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl., 84 (1981), 282-294. doi: 10.1016/0022-247X(81)90166-9.
[38]	G. B. Li, S. S. Yan and J. F. Yang, The super linear 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.
[39]	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.
[40]	C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.
[41]	F. Mahmoudi and A. Malchiodi, Concentration on minimal sub manifolds for a singularly perturbed Neumann problem, Adv. in Math., 209 (2007), 460-525. doi: 10.1016/j.aim.2006.05.014.
[42]	F. Mahmoudi and A. Malchiodi, Concentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 17 (2006), 279-290. doi: 10.4171/RLM/469.
[43]	A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, Geom. Funct. Anal., 15 (2005), 1162-1222. doi: 10.1007/s00039-005-0542-7.
[44]	A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\mathbb{R}^n$, Advances in Mathematics, 221 (2009), 1843-1909. doi: 10.1016/j.aim.2009.03.012.
[45]	A. Malchiodi 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.
[46]	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.
[47]	A. Malchiodi, W.-M. Ni and J. C. Wei, Multiple clustered layer solutions for semi-linear Neumann problems on a ball, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 143-163. doi: 10.1016/j.anihpc.2004.05.003.
[48]	B. B. Manna and S. Santra, On the Hollman McKenna conjecture: Interior concentration On Geodesics, Discrete Contin. Dyn. Syst., 36 (2016), 5595-5626.
[49]	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.
[50]	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.
[51]	R. Mazzeo and F. Pacard, CMC surfaces with Delaunay ends, Comm. Anal. Geom., 9 (2001), 169-237. doi: 10.4310/CAG.2001.v9.n1.a6.
[52]	W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices of Amer. Math. Soc., 45 (1998), 9-18.
[53]	W.-M. Ni and I. Takagi, On the shape of least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851. doi: 10.1002/cpa.3160440705.
[54]	W.-M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4.
[55]	W.-M. Ni and J. C. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768. doi: 10.1002/cpa.3160480704.
[56]	J. C. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Diff. Eqns., 134 (1997), 104-133. doi: 10.1006/jdeq.1996.3218.
[57]	J. C. Wei and S. S. Yan, Lazer-McKenna conjecture: The critical case, J. Funct. Anal., 244 (2007), 639-667. doi: 10.1016/j.jfa.2006.11.002.
[58]	J. C. 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.

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On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem

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On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem

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