UC Irvine

We study the existence of semi-classical bound states of the nonlinear Schrödinger equation \begin{linenomath} -\varepsilon2\Delta u+V(x)u=f(u),\quad x\in {\bf R}^N,\end{linenomath} where N ≥ 3;, Ï is a positive parameter; V:R N → [0, ∞) satisfies some suitable assumptions. We study two cases: if f is asymptotically linear, i.e., if lim|t| → ∞ f(t)/t=constant, then we get positive solutions. If f is superlinear and f(u)=|u| p-2 u+|u| q-2 u, 2* > p > q > 2, we obtain the existence of multiple sign-changing semi-classical bound states with information on the estimates of the energies, the Morse indices and the number of nodal domains. For this purpose, we establish a mountain cliff theorem without the compactness condition and apply a new sign-changing critical point theorem. © 2013 Cambridge Philosophical Society.