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Asymptotics for a generalized Cahn-Hilliard equation with forcing terms
On the location of a peak point of a least energy solution for Hénon equation
1. | Department of Mathematics & PMI, POSTECH, Pohang, Kyungbuk 790-784 |
2. | Department of Mathematics Education, Gwangju National University of Education, 93 Pilmunlo Bugku, Gwangju 500-703 |
3. | Department of Mathematics, POSTECH, Pohang, Kyungbuk 790-784 |
$\Delta u + |x|^{\alpha}u^{p} = 0, \ u > 0 \quad$ in
$\Omega$,
$\ u = 0 \quad$ on $\partial \Omega $,
References:
[1] |
T. Aubin, "Some Nonlinear Problems In Riemannian Geometry,", Springer Monographs in Mathematics, (1998).
|
[2] |
Adimurthi, G. Mancini and S. L. Yadava, The role of the mean curvature in semilinear Neumann problem involving critical exponent,, Comm. in Partial Differential Equations, 20 (1995), 591.
|
[3] |
J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations,, J. Differential Equations, 163 (2000), 429.
|
[4] |
J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations, II,, J. Differential Equations, 173 (2001), 321.
|
[5] |
J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity,, J. Differential Equations, 244 (2008), 2473.
|
[6] |
J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds,, Calc. Var. and Partial Differential Equations, 24 (2005), 459.
|
[7] |
J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground states,, Ann. Inst. H. Poincare Anal. Non Lineaire, 23 (2006), 803.
doi: 10.1016/j.anihpc.2006.04.001. |
[8] |
J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground states II,, J. Differential Equations, 216 (2005), 78.
|
[9] |
D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation,, J. Math. Anal. Appl., 278 (2003), 1.
doi: 10.1016/S0022-247X(02)00292-5. |
[10] |
D. Cao, S. Peng and S. Yan, Asymptotic behaviour of ground state solutions for the Hénon equation,, IMA J. Appl. Math., 74 (2009), 468.
doi: 10.1093/imamat/hxn035. |
[11] |
M. Gazzini and E. Serra, The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues,, Ann. Inst. H. Poincare Anal. Non Lineaire, 25 (2008), 281.
doi: 10.1016/j.anihpc.2006.09.003. |
[12] |
G. Chen, W. M. Ni and J. Zhou, Algorithms and visualization for solutions of nonlinear ellptic equations,, Inter. Jour. Bifur. Chaos, 10 (2000), 1565.
doi: 10.1142/S0218127400001006. |
[13] |
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.
doi: 10.1512/iumj.1999.48.1596. |
[14] |
G. F. D. Duff, Partial differential equations,, in, (1956).
|
[15] |
P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbb R^2$,, J. Anal. Math., 100 (2006), 249.
doi: 10.1007/BF02916763. |
[16] |
B. Gidas, W. N. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.
doi: 10.1007/BF01221125. |
[17] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2nd edition, (1983).
|
[18] |
M. Gruter and K. Widman, The Green function for uniformly elliptic equations,, Manuscripta Math., 37 (1982), 303.
doi: 10.1007/BF01166225. |
[19] |
N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms,, J. Differential Equations, 247 (2009), 1311.
|
[20] |
M. Hénon, Numerical experiments on the spherical stellar systems,, Astronomy and Astrophysics, 24 (1973), 229. Google Scholar |
[21] |
B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Mathematics, 1150 (1985).
|
[22] |
Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations,, Comm. Pure Appl. Math., 51 (1998), 1445.
doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z. |
[23] |
W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications,, Indiana Univ. Math. J., 31(6) (1982), 801.
doi: 10.1512/iumj.1982.31.31056. |
[24] |
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.
doi: 10.1002/cpa.3160440705. |
[25] |
W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247.
doi: 10.1215/S0012-7094-93-07004-4. |
[26] |
W. M. Ni, X. B. Pan and I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents,, Duke Math. J., 67 (1992), 1.
doi: 10.1215/S0012-7094-92-06701-9. |
[27] |
W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, Comm. Pure Appl. Math., 48 (1995), 731.
doi: 10.1002/cpa.3160480704. |
[28] |
A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth,, Math. Z., 256 (2007), 75.
doi: 10.1007/s00209-006-0060-9. |
[29] |
S. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0.$,, Soviet Math. Dokl., 6 (1965), 1408.
|
[30] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984).
|
[31] |
O. Rey, Boundary effect for an elliptic Neumann problem with critical nonlinearity,, Comm. in Partial Differential Equations, 22 (1997), 1055.
|
[32] |
S. Secchi and E. Serra, Symmetry breaking results for problems with exponential growth in the unit disk,, Commun. Contemp. Math., 8 (2006), 823.
doi: 10.1142/S0219199706002295. |
[33] |
E. Serra, Non radial positive solutions for the Hénon equation with critical growth,, Calc. Var. Partial Differential Equations, 23 (2005), 301.
|
[34] |
M. Struwe, "Variational Methods; Application to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Springer-Verlag, (1990).
|
[35] |
D. Smets, J. Su and M. Willem, Nonradial ground states for the Hénon equation,, Communications in Contemporary Mathematics, 4 (2002), 467.
doi: 10.1142/S0219199702000725. |
[36] |
N. Varopoulos, Potential theory and diffusion on Riemannian manifolds, conference on harmonic analysis in honor of Antoni Zygmund,, Wadsworth Math. Ser., (1983), 821.
|
[37] |
J. Wei and S. Yan, Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth,, J. Math. Pures Appl., 88 (2007), 350.
doi: 10.1016/j.matpur.2007.07.001. |
show all references
References:
[1] |
T. Aubin, "Some Nonlinear Problems In Riemannian Geometry,", Springer Monographs in Mathematics, (1998).
|
[2] |
Adimurthi, G. Mancini and S. L. Yadava, The role of the mean curvature in semilinear Neumann problem involving critical exponent,, Comm. in Partial Differential Equations, 20 (1995), 591.
|
[3] |
J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations,, J. Differential Equations, 163 (2000), 429.
|
[4] |
J. Byeon, Effect of symmetry to the structure of positive solutions in nonlinear elliptic equations, II,, J. Differential Equations, 173 (2001), 321.
|
[5] |
J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity,, J. Differential Equations, 244 (2008), 2473.
|
[6] |
J. Byeon and J. Park, Singularly perturbed nonlinear elliptic problems on manifolds,, Calc. Var. and Partial Differential Equations, 24 (2005), 459.
|
[7] |
J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground states,, Ann. Inst. H. Poincare Anal. Non Lineaire, 23 (2006), 803.
doi: 10.1016/j.anihpc.2006.04.001. |
[8] |
J. Byeon and Z. Q. Wang, On the Hénon equation: Asymptotic profile of ground states II,, J. Differential Equations, 216 (2005), 78.
|
[9] |
D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation,, J. Math. Anal. Appl., 278 (2003), 1.
doi: 10.1016/S0022-247X(02)00292-5. |
[10] |
D. Cao, S. Peng and S. Yan, Asymptotic behaviour of ground state solutions for the Hénon equation,, IMA J. Appl. Math., 74 (2009), 468.
doi: 10.1093/imamat/hxn035. |
[11] |
M. Gazzini and E. Serra, The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues,, Ann. Inst. H. Poincare Anal. Non Lineaire, 25 (2008), 281.
doi: 10.1016/j.anihpc.2006.09.003. |
[12] |
G. Chen, W. M. Ni and J. Zhou, Algorithms and visualization for solutions of nonlinear ellptic equations,, Inter. Jour. Bifur. Chaos, 10 (2000), 1565.
doi: 10.1142/S0218127400001006. |
[13] |
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.
doi: 10.1512/iumj.1999.48.1596. |
[14] |
G. F. D. Duff, Partial differential equations,, in, (1956).
|
[15] |
P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbb R^2$,, J. Anal. Math., 100 (2006), 249.
doi: 10.1007/BF02916763. |
[16] |
B. Gidas, W. N. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.
doi: 10.1007/BF01221125. |
[17] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2nd edition, (1983).
|
[18] |
M. Gruter and K. Widman, The Green function for uniformly elliptic equations,, Manuscripta Math., 37 (1982), 303.
doi: 10.1007/BF01166225. |
[19] |
N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms,, J. Differential Equations, 247 (2009), 1311.
|
[20] |
M. Hénon, Numerical experiments on the spherical stellar systems,, Astronomy and Astrophysics, 24 (1973), 229. Google Scholar |
[21] |
B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Mathematics, 1150 (1985).
|
[22] |
Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations,, Comm. Pure Appl. Math., 51 (1998), 1445.
doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z. |
[23] |
W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications,, Indiana Univ. Math. J., 31(6) (1982), 801.
doi: 10.1512/iumj.1982.31.31056. |
[24] |
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.
doi: 10.1002/cpa.3160440705. |
[25] |
W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247.
doi: 10.1215/S0012-7094-93-07004-4. |
[26] |
W. M. Ni, X. B. Pan and I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents,, Duke Math. J., 67 (1992), 1.
doi: 10.1215/S0012-7094-92-06701-9. |
[27] |
W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, Comm. Pure Appl. Math., 48 (1995), 731.
doi: 10.1002/cpa.3160480704. |
[28] |
A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth,, Math. Z., 256 (2007), 75.
doi: 10.1007/s00209-006-0060-9. |
[29] |
S. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0.$,, Soviet Math. Dokl., 6 (1965), 1408.
|
[30] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984).
|
[31] |
O. Rey, Boundary effect for an elliptic Neumann problem with critical nonlinearity,, Comm. in Partial Differential Equations, 22 (1997), 1055.
|
[32] |
S. Secchi and E. Serra, Symmetry breaking results for problems with exponential growth in the unit disk,, Commun. Contemp. Math., 8 (2006), 823.
doi: 10.1142/S0219199706002295. |
[33] |
E. Serra, Non radial positive solutions for the Hénon equation with critical growth,, Calc. Var. Partial Differential Equations, 23 (2005), 301.
|
[34] |
M. Struwe, "Variational Methods; Application to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Springer-Verlag, (1990).
|
[35] |
D. Smets, J. Su and M. Willem, Nonradial ground states for the Hénon equation,, Communications in Contemporary Mathematics, 4 (2002), 467.
doi: 10.1142/S0219199702000725. |
[36] |
N. Varopoulos, Potential theory and diffusion on Riemannian manifolds, conference on harmonic analysis in honor of Antoni Zygmund,, Wadsworth Math. Ser., (1983), 821.
|
[37] |
J. Wei and S. Yan, Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth,, J. Math. Pures Appl., 88 (2007), 350.
doi: 10.1016/j.matpur.2007.07.001. |
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