Two nonlinear days in Urbino 2017. Electron. J. Diff. Eqns., Conference 25 (2018), pp. 39-53.

Ground states of some coupled nonlocal fractional dispersive PDEs

Eduardo Colorado

We show the existence of ground state solutions to the following stationary system coming from some coupled fractional dispersive equations such as: nonlinear fractional Schrodinger (NLFS) equations (for dimension n=1,2,3) or NLFS and fractional Korteweg-de Vries equations (for n=1),
 (-\Delta)^{s} u+ \lambda_1 u =  u^{3}+\beta uv,\quad 
  u\in  W^{s,2}(\mathbb{R}^n)\cr
 (-\Delta)^{s} v+ \lambda_2 v =  \frac 12 v^{2}+\frac 12 \beta u^2,\quad 
 v\in  W^{s,2}(\mathbb{R}^n),
where $\lambda_j>0$, j=1,2, $\beta\in \mathbb{R}$, n=1,2,3, and n/4 < s < 1. Precisely, we prove the existence of a positive radially symmetric ground state for any $\beta>0$.

Published September 15, 2018.
Math Subject Classifications: 49J40, 35Q55, 35Q53, 35B38, 35J50.
Key Words: Nonlinear Fractional Schrodinger equation; variational method; fractional Korteweg-de Vries equation; critical point theory; ground state

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Eduardo Colorado
Departamento de Matemáticas
Universidad Carlos III de Madrid
Avda. Universidad 30, 28911 Leganés
Madrid, Spain

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