Nonlinear Schrodinger equations with potentials vanishing at infinity

Bonheure, Denis [UCL] Van Schaftingen, Jean [UCL]

In this Note, we deal with stationary nonlinear Schrodinger equations of the form epsilon(2)Delta u + V(x)u = K(x)u(p), x is an element of R-N, where V, K > 0 and p > 1 is subcritical. We allow the potential V to vanish at infinity and the competing function K to be unbounded. In this framework, positive ground states may not exist. We prove the existence of at least one positive bound state solution in the semi-classical limit, i.e. for epsilon similar to 0. We also investigate the qualitative properties of the solution as 0.