Solitary Waves for Some Nonlinear Schrödinger Systems

In this paper we study the existence of radially symmetric positive solutions in

$$ H_{\text{rad}}^{1} ({\mathbb{R}}^{N} )\; \times \;H_{\text{rad}}^{1} ({\mathbb{R}}^{N} ) $$

of the elliptic system:

$$ - \Updelta u + u - (\alpha u^{2} + \beta v^{2} )u\, = \,0, $$

$$ - \Updelta v + \omega^{2} v - (\beta u^{2} + \gamma v^{2} )v\, = \,0, $$

N = 1, 2, 3, where

α

and

γ

are positive constants ( will be allowed to be negative). This system has trivial solutions of the form (

ϕ

, 0) and (0,) where

ϕ

and are nontrivial solutions of scalar equations. The existence of nontrivial solutions for some values of the parameters

α

,

β

,

γ

,

ω

has been studied recently by several authors [A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453–458; T.C. Lin, J. Wei, Ground states of

N

coupled nonlinear Schrödinger equations in

R

n

,

n

≤ 3, Comm. Math. Phys. 255 (2005) 629–653; T.C. Lin, J. Wei, Ground states of

N

coupled nonlinear Schrödinger equations in

R

n

,

n

≤ 3, Comm. Math. Phys., Erratum, in press; L. Maia, E.Montefusco, B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, preprint; B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in

R

N

, preprint; J. Yang, Classification of the solitary waves in coupled nonlinear Schrödinger equations, Physica D 108 (1997) 92–112]. For

N

= 2, 3, perhaps the most general existence result has been proved in [A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453–458] under conditions which are equivalent to ours. Motivated by some numerical computations, we return to this problem and, using our approach, we give a more detailed description of the regions of parameters for which existence can be proved. In particular, based also on numerical evidence, we show that the shape of the region of the parameters for which existence of solution can be proved, changes drastically when we pass from dimensions

N

= 1, 2 to dimension

N

= 3. Our approach differs from the ones used before. It relies heavily on the spectral theory for linear elliptic operators. Furthermore, we also consider the case

N

= 1 which has to be treated more extensively due to some lack of compactness for even functions. This case has not been treated before.