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Li-Yorke Chaos for ultragraph shift spaces
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 |
$ \Omega \subset \mathbb {R}^n $ |
$ \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. $ |
$ p $ |
$ p > 1 $ |
$ n = 2 $ |
$ 1 < p < \frac{n+2}{n-2} $ |
$ n \geq 3 $ |
$ \sigma > 0 $ |
$ \nu $ |
$ \partial\Omega $ |
$ \Gamma $ |
$ \partial\Omega $ |
$ \Gamma $ |
$ \Gamma $ |
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. Google Scholar |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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.
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