The Cauchy problem for logarithmic Schr{ö}dinger equation with a perturbation of power law nonlinearity

RESUMO

In this paper, we study the Cauchy problem for the nonlinear Schr{ö}dinger equation with a nonlinearity combining a logarithmic term and a local nonlinearity which has polynomial growth. Our main result is the construction of solutions in the energy space $W_1(R^d)$ in the subcritical case. We also discuss a sufficient condition for global existence in the $L^2$-critical case in each of these spaces. Moreover, we establish a more general Cauchy theory in the space $\Sigma_\alpha$, as introduced without details in [6].