Following the original approach introduced by T. Cazenave and P.L. Lions in \cite{ CaLi} we prove the existence and the orbital stability of standing waves for the following class of NLS: \label{ intr1} i\partial_t u+ \Delta u - V(x) u + Q(x) u|u|^{ p-2}=0, \hbox{ } (t,x) \in \R\times \R^n, \hbox{ } 2<p<2+\frac 4n and \label{ intr2} i\partial_t u - \Delta^2 u - V(x) u + Q(x) u|u|^{ p-2}=0, \hbox{ } (t,x) \in \R\times \R^n, \hbox{ } 2<p<2+\frac 8n under suitable assumptions on the potentials $V(x)$ and $Q(x)$. More precisely we assume $V(x), Q(x) \in L^\infty(\R^n)$ and $meas\{ Q(x)>\lambda_0\}\in (0,\infty)$ for a suitable $\lambda_0>0$. The main point is the analysis of the compactness of minimiziang sequences to suitable constrained minimization problems related to \eqref{ intr1} and \eqref{ intr2}.
Categories: math-ph math.MP