    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

 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

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

##### References:
  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.  doi: 10.1090/S0002-9947-07-04397-8.  Google Scholar  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.  doi: 10.1090/S0002-9947-2011-05292-X.  Google Scholar  F. HadjSelem, Radial solutions with prescribed numbers of zeros for the nonlinear Schrödinger equation with harmonic potential,, Nonlinearity, 24 (2011), 1795.  doi: 10.1088/0951-7715/24/6/006.  Google Scholar  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.  doi: 10.1016/j.jmaa.2011.09.034.  Google Scholar  M. Hirose and M. Ohta, Structure of positive radial solutions to scalar field equations with harmonic potential,, J. Differential Equations, 178 (2002), 519.  doi: 10.1006/jdeq.2000.4010.  Google Scholar  M. Hirose and M. Ohta, Uniqueness of positive solutions to scalar field equations with harmonic potential,, Funkcial. Ekvac., 50 (2007), 67.  doi: 10.1619/fesi.50.67.  Google Scholar  D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (): 241. Google Scholar  Y. Li and W.-M. Ni, Radial symmetry of positive solution of nonlinear elliptic equations in $\mathbbR^N$,, Comm. Partial Differential Equations, 18 (1993), 1043.  doi: 10.1080/03605309308820960.  Google Scholar  F. Merle and L. A. Peletier, Positive solutions of elliptic equations involving supercritical growth,, Proceeding of the Royal Society of Edingburgh, 118A (1991), 49.  doi: 10.1017/S0308210500028882.  Google Scholar  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 (1986), 6.   Google Scholar  X. Wang, On the Cauchy problem for reaction-diffusion equations,, Trans. Amer. Math. Soc., 337 (1993), 549.  doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar
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