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

Abstract:
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),
$$\displaylines{ (-\Delta)^{s} u+ \lambda_1 u = u^{3}+\beta uv,\quad u\in W^{s,2}(\mathbb{R}^n)\cr (-\Delta)^{s} v+ \lambda_2 v = \frac 12 v^{2}+\frac 12 \beta u^2,\quad v\in W^{s,2}(\mathbb{R}^n), }$$
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
email: eduardo.colorado@uc3m.es, eduardo.colorado@icmat.es

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