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Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity
1. | Laboratoire de Mathématiques UMR 6620 - CNRS, Université Blaise Pascal, Campus des Cézeaux -B.P. 80026, 63171 Aubière cedex, France |
2. | Department of Mathematics, Tsuda College, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo 187-8577, Japan |
3. | Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong |
References:
[1] |
J. Dolbeault and I. Flores, Geometry of phase space and solutions of semilinear elliptic equations in a ball, Trans. Amer. Math. Soc., 359 (2007), 4073-4087.
doi: 10.1090/S0002-9947-07-04397-8. |
[2] |
Z. Guo and J. Wei, Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent, Trans. Amer. Math. Soc., 363 (2011), 4777-4799.
doi: 10.1090/S0002-9947-2011-05292-X. |
[3] |
F. HadjSelem, Radial solutions with prescribed numbers of zeros for the nonlinear Schrödinger equation with harmonic potential, Nonlinearity, 24 (2011), 1795-1819.
doi: 10.1088/0951-7715/24/6/006. |
[4] |
F. HadjSelem and H. Kikuchi, Existence and non-existence of solution for semilinear elliptic equation with harmonic potential and Sobolev critical/supercritical nonlinearities, J. Math. Anal. Appl., 387 (2012), 746-754.
doi: 10.1016/j.jmaa.2011.09.034. |
[5] |
M. Hirose and M. Ohta, Structure of positive radial solutions to scalar field equations with harmonic potential, J. Differential Equations, 178 (2002), 519-540.
doi: 10.1006/jdeq.2000.4010. |
[6] |
M. Hirose and M. Ohta, Uniqueness of positive solutions to scalar field equations with harmonic potential, Funkcial. Ekvac., 50 (2007), 67-100.
doi: 10.1619/fesi.50.67. |
[7] |
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269. |
[8] |
Y. Li and W.-M. Ni, Radial symmetry of positive solution of nonlinear elliptic equations in $\mathbb{R}^N2$, Comm. Partial Differential Equations, 18 (1993), 1043-1054.
doi: 10.1080/03605309308820960. |
[9] |
F. Merle and L. A. Peletier, Positive solutions of elliptic equations involving supercritical growth, Proceeding of the Royal Society of Edingburgh, 118A (1991), 49-62.
doi: 10.1017/S0308210500028882. |
[10] |
W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The aumalous case, Atti dei Convegni Lincei, 77, Roma, 6-9 maggio (1986). |
[11] |
X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.
doi: 10.1090/S0002-9947-1993-1153016-5. |
show all references
References:
[1] |
J. Dolbeault and I. Flores, Geometry of phase space and solutions of semilinear elliptic equations in a ball, Trans. Amer. Math. Soc., 359 (2007), 4073-4087.
doi: 10.1090/S0002-9947-07-04397-8. |
[2] |
Z. Guo and J. Wei, Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent, Trans. Amer. Math. Soc., 363 (2011), 4777-4799.
doi: 10.1090/S0002-9947-2011-05292-X. |
[3] |
F. HadjSelem, Radial solutions with prescribed numbers of zeros for the nonlinear Schrödinger equation with harmonic potential, Nonlinearity, 24 (2011), 1795-1819.
doi: 10.1088/0951-7715/24/6/006. |
[4] |
F. HadjSelem and H. Kikuchi, Existence and non-existence of solution for semilinear elliptic equation with harmonic potential and Sobolev critical/supercritical nonlinearities, J. Math. Anal. Appl., 387 (2012), 746-754.
doi: 10.1016/j.jmaa.2011.09.034. |
[5] |
M. Hirose and M. Ohta, Structure of positive radial solutions to scalar field equations with harmonic potential, J. Differential Equations, 178 (2002), 519-540.
doi: 10.1006/jdeq.2000.4010. |
[6] |
M. Hirose and M. Ohta, Uniqueness of positive solutions to scalar field equations with harmonic potential, Funkcial. Ekvac., 50 (2007), 67-100.
doi: 10.1619/fesi.50.67. |
[7] |
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269. |
[8] |
Y. Li and W.-M. Ni, Radial symmetry of positive solution of nonlinear elliptic equations in $\mathbb{R}^N2$, Comm. Partial Differential Equations, 18 (1993), 1043-1054.
doi: 10.1080/03605309308820960. |
[9] |
F. Merle and L. A. Peletier, Positive solutions of elliptic equations involving supercritical growth, Proceeding of the Royal Society of Edingburgh, 118A (1991), 49-62.
doi: 10.1017/S0308210500028882. |
[10] |
W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The aumalous case, Atti dei Convegni Lincei, 77, Roma, 6-9 maggio (1986). |
[11] |
X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590.
doi: 10.1090/S0002-9947-1993-1153016-5. |
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