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Quasilinear retarded differential equations with functional dependence on piecewise constant argument
Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space
1. | Department of Mathematics, National Central University, Chung-Li, 32001, Taiwan |
2. | Department of Mathematics, National Central University, Chung-Li 32001 |
3. | Department of Mathematical Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, 599-8531 |
References:
[1] |
S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $R^n$, J. Differential Equations, 194 (2003), 460-499.
doi: 10.1006/jdeq.2001.4162. |
[2] |
S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $R^n$, J. Differential Equations, 185 (2002), 225-250.
doi: 10.1016/s0022-396(03)00172-4. |
[3] |
C. Bandle, A. Brillard and M. Flucher, Green's function, harmonic transplantation and best Sobolev constant in spaces of constant curvature, Trans. Amer. Math. Soc., 350 (1998), 1103-1128.
doi: 10.1090/S0002-9947-98-02085-6. |
[4] |
C. Bandle and Y. Kabeya, On the positive, "radial" solutions of a semilinear elliptic equation on $H^N$, Adv. Nonlinear Anal., 1 (2012), 1-25.
doi: 10.1515/ana-2011-0004. |
[5] |
C. Bandle and M. Marcus, The positive radial solutions of a class of semilinear elliptic equations, J. Reine Angew. Math., 401 (1989), 25-59. |
[6] |
M. Bonforte, F. Gazzola, G. Grillo and J. L. Vazquez, Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space, Cal. Var. Partial Differential Equations, 46 (2013), 375-401. |
[7] |
J.-L. Chern, Z.-Y. Chen, J-H. Chen and Y.-L. Tang, On the classification of standing wave solutions for the Schrödinger equation, Comm. Partial Differential Equations, 35 (2010), 275-301;
doi: 10.1080/03605300903419767. |
[8] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[9] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Adv. Math. Suppl. Stud., 7A (1981), 369-402. |
[10] |
A. Grigor'yan, "Heat Kernel and Analysis on Manifolds'', AMS, Providence, 2009. |
[11] |
C. Gui, W.-M. Ni and X.-F. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $R^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.
doi: 10.1002/cpa.3160450906. |
[12] |
P. Hartman, "Ordinary Differential Equations'', Birkhäuser, Boston, second edition, 1982. |
[13] |
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269. |
[14] |
S. Kumaresan and J. Prajapat, Analogue of Gidas-Ni-Nirenberg result in hyperbolic space and sphere, Rend. Inst. Math. Univ. Trieste, 30 (1998), 107-112. |
[15] |
Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p = 0$ in $R^n$, J. Differential Equations, 95 (1992), 304-330.
doi: 10.1016/0022-0396(92)90034-K. |
[16] |
Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406.
doi: 10.1006/jdeq.1999.3735. |
[17] |
G. Mancini and K. Sandeep, On a semilinear elliptic equation in $H^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (2008), 635-671. |
[18] |
W.-M. Ni, On the positive radial solutions of some semilinear elliptic equations on $R^n$, Appl. Math. Optim., 9 (1983), 373-380. |
[19] |
W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math., 5 (1988), 1-32. |
[20] |
S. Stapelkamp, The Brezis-Nirenberg problem on $H^n$: existence and uniqueness of solutions, in "Elliptic and Parabolic Problems- Rolduc and Gaeta 2001,'' Bemelmans et al. ed., World Scientific Publ. River Edge, NJ, (2002), 283-290. |
[21] |
X.-F. Wang, On Cauchy Problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.
doi: 10.1090/S0002-9947-1993-1153015-5. |
show all references
References:
[1] |
S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $R^n$, J. Differential Equations, 194 (2003), 460-499.
doi: 10.1006/jdeq.2001.4162. |
[2] |
S. Bae and T. K. Chang, On a class of semilinear elliptic equations in $R^n$, J. Differential Equations, 185 (2002), 225-250.
doi: 10.1016/s0022-396(03)00172-4. |
[3] |
C. Bandle, A. Brillard and M. Flucher, Green's function, harmonic transplantation and best Sobolev constant in spaces of constant curvature, Trans. Amer. Math. Soc., 350 (1998), 1103-1128.
doi: 10.1090/S0002-9947-98-02085-6. |
[4] |
C. Bandle and Y. Kabeya, On the positive, "radial" solutions of a semilinear elliptic equation on $H^N$, Adv. Nonlinear Anal., 1 (2012), 1-25.
doi: 10.1515/ana-2011-0004. |
[5] |
C. Bandle and M. Marcus, The positive radial solutions of a class of semilinear elliptic equations, J. Reine Angew. Math., 401 (1989), 25-59. |
[6] |
M. Bonforte, F. Gazzola, G. Grillo and J. L. Vazquez, Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space, Cal. Var. Partial Differential Equations, 46 (2013), 375-401. |
[7] |
J.-L. Chern, Z.-Y. Chen, J-H. Chen and Y.-L. Tang, On the classification of standing wave solutions for the Schrödinger equation, Comm. Partial Differential Equations, 35 (2010), 275-301;
doi: 10.1080/03605300903419767. |
[8] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[9] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Adv. Math. Suppl. Stud., 7A (1981), 369-402. |
[10] |
A. Grigor'yan, "Heat Kernel and Analysis on Manifolds'', AMS, Providence, 2009. |
[11] |
C. Gui, W.-M. Ni and X.-F. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $R^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.
doi: 10.1002/cpa.3160450906. |
[12] |
P. Hartman, "Ordinary Differential Equations'', Birkhäuser, Boston, second edition, 1982. |
[13] |
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269. |
[14] |
S. Kumaresan and J. Prajapat, Analogue of Gidas-Ni-Nirenberg result in hyperbolic space and sphere, Rend. Inst. Math. Univ. Trieste, 30 (1998), 107-112. |
[15] |
Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u + K(x)u^p = 0$ in $R^n$, J. Differential Equations, 95 (1992), 304-330.
doi: 10.1016/0022-0396(92)90034-K. |
[16] |
Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406.
doi: 10.1006/jdeq.1999.3735. |
[17] |
G. Mancini and K. Sandeep, On a semilinear elliptic equation in $H^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (2008), 635-671. |
[18] |
W.-M. Ni, On the positive radial solutions of some semilinear elliptic equations on $R^n$, Appl. Math. Optim., 9 (1983), 373-380. |
[19] |
W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math., 5 (1988), 1-32. |
[20] |
S. Stapelkamp, The Brezis-Nirenberg problem on $H^n$: existence and uniqueness of solutions, in "Elliptic and Parabolic Problems- Rolduc and Gaeta 2001,'' Bemelmans et al. ed., World Scientific Publ. River Edge, NJ, (2002), 283-290. |
[21] |
X.-F. Wang, On Cauchy Problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.
doi: 10.1090/S0002-9947-1993-1153015-5. |
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