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 - A, 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

[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]

A. MalchiodiW.-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.  Google Scholar

[48]

B. B. Manna and S. Santra, On the Hollman McKenna conjecture: Interior concentration On Geodesics, Discrete Contin. Dyn. Syst., 36 (2016), 5595-5626.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices of Amer. Math. Soc., 45 (1998), 9-18.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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]

A. MalchiodiW.-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.  Google Scholar

[48]

B. B. Manna and S. Santra, On the Hollman McKenna conjecture: Interior concentration On Geodesics, Discrete Contin. Dyn. Syst., 36 (2016), 5595-5626.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[52]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices of Amer. Math. Soc., 45 (1998), 9-18.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

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

[2]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

[3]

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

[4]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739

[5]

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

[6]

Shuangjie Peng, Jing Zhou. Concentration of solutions for a Paneitz type problem. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1055-1072. doi: 10.3934/dcds.2010.26.1055

[7]

Jun Wang, Lu Xiao. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7137-7168. doi: 10.3934/dcds.2016111

[8]

Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465

[9]

Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial & Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259

[10]

Julii A. Dubinskii. Complex Neumann type boundary problem and decomposition of Lebesgue spaces. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 201-210. doi: 10.3934/dcds.2004.10.201

[11]

Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671

[12]

Zhongyuan Liu. Concentration of solutions for the fractional Nirenberg problem. Communications on Pure & Applied Analysis, 2016, 15 (2) : 563-576. doi: 10.3934/cpaa.2016.15.563

[13]

Vincenzo Ambrosio. Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2265-2284. doi: 10.3934/dcds.2017099

[14]

Antonia Chinnì, Roberto Livrea. Multiple solutions for a Neumann-type differential inclusion problem involving the $p(\cdot)$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 753-764. doi: 10.3934/dcdss.2012.5.753

[15]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[16]

Serafin Bautista, Yeison Sánchez. Sectional-hyperbolic Lyapunov stable sets. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2011-2016. doi: 10.3934/dcds.2020103

[17]

Peter Benner, Tobias Breiten, Carsten Hartmann, Burkhard Schmidt. Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations. Journal of Computational Dynamics, 2020, 7 (1) : 1-33. doi: 10.3934/jcd.2020001

[18]

Dimitri Breda, Sara Della Schiava. Pseudospectral reduction to compute Lyapunov exponents of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2727-2741. doi: 10.3934/dcdsb.2018092

[19]

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

[20]

Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (118)
  • HTML views (160)
  • Cited by (0)

Other articles
by authors

[Back to Top]