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July  2019, 18(4): 1921-1965. doi: 10.3934/cpaa.2019089

Spike layer solutions for a singularly perturbed Neumann problem: Variational construction without a nondegeneracy

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

* Corresponding author

Received  August 2018 Revised  August 2018 Published  January 2019

Fund Project: The first author is supported by NRF-2017R1A2B4007816

We consider the following singularly perturbed problem
$ \begin{equation*} \varepsilon ^2 \Delta u - u + f(u) = 0,\ \ \ \, u>0 \text{ in } \Omega, \ \ \ \ \frac{\partial u}{\partial \nu} = 0 \text{ on } \partial\Omega. \end{equation*} $
Existence of a solution with a spike layer near a min-max critical point of the mean curvature on the boundary
$ \partial \Omega $
is well known when a nondegeneracy for a limiting problem holds. In this paper, we use a variational method for the construction of such a solution which does not depend on the nondengeneracy for the limiting problem. By a purely variational approach, we construct the solution for an optimal class of nonlinearities
$ f $
satisfying the Berestycki-Lions conditions.
Citation: Jaeyoung Byeon, Sang-hyuck Moon. Spike layer solutions for a singularly perturbed Neumann problem: Variational construction without a nondegeneracy. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1921-1965. doi: 10.3934/cpaa.2019089
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[3]

P. BatesK. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433.  doi: 10.1007/s00222-008-0141-y.  Google Scholar

[4]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/bf00250555.  Google Scholar

[5]

J. Byeon, Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains, Comm. Partial Differential Equations, 22 (1997), 1731-1769.  doi: 10.1080/03605309708821317.  Google Scholar

[6]

J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity, J. Differential Equations, 244 (2008), 2473-2497.  doi: 10.1016/j.jde.2008.02.024.  Google Scholar

[7]

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200.  doi: 10.1007/s00205-006-0019-3.  Google Scholar

[8]

J. Byeon and Y. Lee, Singularly perturbed nonlinear Neumann problems under the conditions of Berestycki and Lions, J. Differential Equations, 252 (2012), 3848-3872.  doi: 10.1016/j.jde.2011.12.013.  Google Scholar

[9]

J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds, Calc. Var. Partial Differential Equations, 24 (2005), 459-477.  doi: 10.1007/s00526-005-0339-4.  Google Scholar

[10]

J. Byeon and K. Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential, J. Eur. Math. Soc., 15 (2013), 1859-1899.  doi: 10.4171/jems/407.  Google Scholar

[11]

J. Byeon and K. Tanaka, Multi-bump positive solutions for a nonlinear elliptic problem in expanding tubular domains, Calc. Var. Partial Differential Equations, 50 (2014), 365-397.  doi: 10.1007/s00526-013-0639-z.  Google Scholar

[12]

C. CortázarM. Elgueta and P. Felmer, Uniqueness of positive solutions of $Δu+f(u) = 0$ in $ \mathbb {R} ^N, N \le 3$, Arch. Ration. Mech. Anal., 142 (1998), 127-141.  doi: 10.1007/s002050050086.  Google Scholar

[13]

V. Coti Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $ \mathbb {R} ^N$, Comm. Pure Appl. Math., 45 (1992), 1217-1269.  doi: 10.1002/cpa.3160451002.  Google Scholar

[14]

E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262.  doi: 10.2140/pjm.1999.189.241.  Google Scholar

[15]

J. DávilaM. del Pino and I. Guerra, Non-uniqueness of positive ground states of non-linear Schrödinger equations, Proc. Lond. Math. Soc., 106 (2013), 318-344.  doi: 10.1112/plms/pds038.  Google Scholar

[16]

M. del Pino and P. L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999), 883-898.  doi: 10.1512/iumj.1999.48.1596.  Google Scholar

[17]

M. del PinoP. L. Felmer and J. 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

[18]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar

[19]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.   Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^nd$ edition, Springer-Verlag, Berlin, 1983.  Google Scholar

[21]

C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769.  doi: 10.1215/S0012-7094-96-08423-9.  Google Scholar

[22]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $ \mathbb {R} ^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[23]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[24]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $ \mathbb {R} ^n$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[25]

Y. Lee and J. Seok, Multiple interior and boundary peak solutions to singularly perturbed nonlinear Neumann problems under the Berestycki–Lions condition, Math. Ann., 367 (2017), 881-928.  doi: 10.1007/s00208-016-1412-3.  Google Scholar

[26]

Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations, 21 (1998), 487-545.  doi: 10.1080/03605309808821354.  Google Scholar

[27]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case, part Ⅱ, Ann. Inst. H. Poincaré, 1 (1984), 223-283.   Google Scholar

[28]

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

[29]

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

[30]

F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math., 209 (2007), 460-525.  doi: 10.1016/j.aim.2006.05.014.  Google Scholar

[31]

A. Malchiodi, Construction of multidimensional spike-layers, Discrete Contin. Dyn. Syst., 14 (2006), 187-202.  doi: 10.3934/dcds.2006.14.187.  Google Scholar

[32]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.  doi: 10.1215/S0012-7094-04-12414-5.  Google Scholar

[33]

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

[34]

W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar

[35]

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

[36]

A. Turing, The chemical basis of morphogenesis, Philos. Trans. Royal Soc. (B), 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[37]

J. C. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Differential Equations, 134 (1997), 104-133.  doi: 10.1006/jdeq.1996.3218.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[3]

P. BatesK. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433.  doi: 10.1007/s00222-008-0141-y.  Google Scholar

[4]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/bf00250555.  Google Scholar

[5]

J. Byeon, Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains, Comm. Partial Differential Equations, 22 (1997), 1731-1769.  doi: 10.1080/03605309708821317.  Google Scholar

[6]

J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity, J. Differential Equations, 244 (2008), 2473-2497.  doi: 10.1016/j.jde.2008.02.024.  Google Scholar

[7]

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200.  doi: 10.1007/s00205-006-0019-3.  Google Scholar

[8]

J. Byeon and Y. Lee, Singularly perturbed nonlinear Neumann problems under the conditions of Berestycki and Lions, J. Differential Equations, 252 (2012), 3848-3872.  doi: 10.1016/j.jde.2011.12.013.  Google Scholar

[9]

J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds, Calc. Var. Partial Differential Equations, 24 (2005), 459-477.  doi: 10.1007/s00526-005-0339-4.  Google Scholar

[10]

J. Byeon and K. Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential, J. Eur. Math. Soc., 15 (2013), 1859-1899.  doi: 10.4171/jems/407.  Google Scholar

[11]

J. Byeon and K. Tanaka, Multi-bump positive solutions for a nonlinear elliptic problem in expanding tubular domains, Calc. Var. Partial Differential Equations, 50 (2014), 365-397.  doi: 10.1007/s00526-013-0639-z.  Google Scholar

[12]

C. CortázarM. Elgueta and P. Felmer, Uniqueness of positive solutions of $Δu+f(u) = 0$ in $ \mathbb {R} ^N, N \le 3$, Arch. Ration. Mech. Anal., 142 (1998), 127-141.  doi: 10.1007/s002050050086.  Google Scholar

[13]

V. Coti Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $ \mathbb {R} ^N$, Comm. Pure Appl. Math., 45 (1992), 1217-1269.  doi: 10.1002/cpa.3160451002.  Google Scholar

[14]

E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262.  doi: 10.2140/pjm.1999.189.241.  Google Scholar

[15]

J. DávilaM. del Pino and I. Guerra, Non-uniqueness of positive ground states of non-linear Schrödinger equations, Proc. Lond. Math. Soc., 106 (2013), 318-344.  doi: 10.1112/plms/pds038.  Google Scholar

[16]

M. del Pino and P. L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999), 883-898.  doi: 10.1512/iumj.1999.48.1596.  Google Scholar

[17]

M. del PinoP. L. Felmer and J. 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

[18]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar

[19]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.   Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^nd$ edition, Springer-Verlag, Berlin, 1983.  Google Scholar

[21]

C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769.  doi: 10.1215/S0012-7094-96-08423-9.  Google Scholar

[22]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $ \mathbb {R} ^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[23]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[24]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u - u + u^p=0$ in $ \mathbb {R} ^n$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[25]

Y. Lee and J. Seok, Multiple interior and boundary peak solutions to singularly perturbed nonlinear Neumann problems under the Berestycki–Lions condition, Math. Ann., 367 (2017), 881-928.  doi: 10.1007/s00208-016-1412-3.  Google Scholar

[26]

Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations, 21 (1998), 487-545.  doi: 10.1080/03605309808821354.  Google Scholar

[27]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case, part Ⅱ, Ann. Inst. H. Poincaré, 1 (1984), 223-283.   Google Scholar

[28]

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

[29]

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

[30]

F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math., 209 (2007), 460-525.  doi: 10.1016/j.aim.2006.05.014.  Google Scholar

[31]

A. Malchiodi, Construction of multidimensional spike-layers, Discrete Contin. Dyn. Syst., 14 (2006), 187-202.  doi: 10.3934/dcds.2006.14.187.  Google Scholar

[32]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.  doi: 10.1215/S0012-7094-04-12414-5.  Google Scholar

[33]

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

[34]

W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar

[35]

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

[36]

A. Turing, The chemical basis of morphogenesis, Philos. Trans. Royal Soc. (B), 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[37]

J. C. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Differential Equations, 134 (1997), 104-133.  doi: 10.1006/jdeq.1996.3218.  Google Scholar

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