Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity

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October 2013, 33(10): 4613-4626. doi: 10.3934/dcds.2013.33.4613

Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity

Fouad Hadj Selem ^1, , Hiroaki Kikuchi ^2, and Juncheng Wei ^3,

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

Received December 2012 Revised March 2013 Published April 2013

Abstract
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In this paper, we study a singular solution to the following elliptic equations: \begin{equation*} \left\{\begin{array}{ll} - \Delta u + |x|^{2}u - \lambda u - |u|^{p-1}u = 0, \quad x \in \mathbb{R}^{d}, & \\ u(x) > 0, \quad x \in \mathbb{R}^{d}, & \\ u(x) \to 0 \quad \text{as}\; |x| \to \infty, & \end{array}\right. \end{equation*} where $d \geq 3, \lambda >0$ and $p > 1$. In the spirit of Merle and Peletier [9], we shall show that in case of $p>(d+2)/(d-2)$, there is a unique value $\lambda = \lambda_{*}$ such that the equation with $\lambda = \lambda_{*}$ has a unique radial singular solution.

Keywords: Sobolev supercritical, singular solution, harmonic potential..

Mathematics Subject Classification: Primary: 35J15, 35J61, 35J91; Secondary: 58F1.

Citation: Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613

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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