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

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

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 $
if
$ n = 2 $
and
$ 1 < p < \frac{n+2}{n-2} $
if
$ 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.
Citation: Imene Bendahou, Zied Khemiri, Fethi Mahmoudi. On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem. Discrete & 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.  Google Scholar

[2]

A. AmbrosettiA. 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.  Google Scholar

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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.  Google Scholar

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W. W. AoM. 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.  Google Scholar

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W. W. AoM. 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.  Google Scholar

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W. W. AoJ. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: The nonhomogeneous case, Adv. Differential Equations, 12 (2007), 961-993.   Google Scholar

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M. del PinoP. 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.  Google Scholar

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M. del PinoP. 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.  Google Scholar

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M. del PinoP. 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.  Google Scholar

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M. del PinoF. 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.  Google Scholar

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M. del PinoM. 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.  Google Scholar

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S. B. DengF. 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.  Google Scholar

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S. B. DengM. 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.  Google Scholar

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A. Gierer and H.Meinhardt, A thoery of biological pattern formation, Kybernetik, 12 (1972), 30–39, http://dx.doi.org/10.1007/BF00289234. Google Scholar

[27]

M. GrossiA. 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.  Google Scholar

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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.  Google Scholar

[29]

C.-F. GuiJ. 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.  Google Scholar

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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.  Google Scholar

[31]

P. Hess and B. Ruf, On a superlinear elliptic boundary value problem, Math. Z., 164 (1978), 9-14.  doi: 10.1007/BF01214785.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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F.-H. LinW.-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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[2]

A. AmbrosettiA. 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.  Google Scholar

[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.  Google Scholar

[4]

A. AmbrosettiE. 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.  Google Scholar

[5]

W. W. AoM. 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.  Google Scholar

[6]

W. W. AoM. 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.  Google Scholar

[7]

W. W. AoJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: The nonhomogeneous case, Adv. Differential Equations, 12 (2007), 961-993.   Google Scholar

[18]

M. del PinoP. 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.  Google Scholar

[19]

M. del PinoP. 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.  Google Scholar

[20]

M. del PinoP. 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.  Google Scholar

[21]

M. del PinoF. 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.  Google Scholar

[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.  Google Scholar

[23]

M. del PinoM. 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.  Google Scholar

[24]

S. B. DengF. 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.  Google Scholar

[25]

S. B. DengM. 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.  Google Scholar

[26]

A. Gierer and H.Meinhardt, A thoery of biological pattern formation, Kybernetik, 12 (1972), 30–39, http://dx.doi.org/10.1007/BF00289234. Google Scholar

[27]

M. GrossiA. 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.  Google Scholar

[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.  Google Scholar

[29]

C.-F. GuiJ. 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.  Google Scholar

[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.  Google Scholar

[31]

P. Hess and B. Ruf, On a superlinear elliptic boundary value problem, Math. Z., 164 (1978), 9-14.  doi: 10.1007/BF01214785.  Google Scholar

[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.  Google Scholar

[33]

N. Kapouleas, Complete CMC surfaces in Euclidean three-space, Ann. of Math., 131 (1990), 239-330.  doi: 10.2307/1971494.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[38]

G. B. LiS. 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.  Google Scholar

[39]

F.-H. LinW.-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.  Google Scholar

[40]

C.-S. LinW.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

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