CONSTRUCTION OF SOLUTIONS FOR SOME LOCALIZED NONLINEAR SCHRODINGER EQUATIONS

For an N-body system of linear Schrodinger equation with space dependent interaction between particles, one would expect that the corresponding one body equation, arising as a mean field approximation, would have a space dependent nonlinearity. With such motivation we consider the following model of a nonlinear reduced Schrodinger equation with space dependent nonlinearity -phi '' + V(x)h'(vertical bar phi vertical bar(2))phi = lambda phi, where V(x) = - chi([-1,1])(x) is minus the characteristic function of the interval [-1, 1] and where h' is any continuous strictly increasing function. In this article, for any negative value of lambda we present the construction and analysis of the infinitely many solutions of this equation, which are localized in space and hence correspond to bound-states of the associated time-dependent version of the equation.