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- Title
Semi-classical bound states of Schrödinger equations.
- Authors
SCHECHTER, M.; ZOU, W.
- Abstract
We study the existence of semi-classical bound states of the nonlinear Schrödinger equation \begin{linenomath}$$ -\varepsilon^2\Delta u+V(x)u=f(u),\quad x\in {\bf R}^N,$$\end{linenomath} where N ≥ 3;, ϵ is a positive parameter; V:RN → [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−2u+|u|q−2u, 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.
- Subjects
BOUND states; NONLINEAR analysis; SEMI-classical model (Atomic physics); COMPACT spaces (Topology); MATHEMATICAL models; LINEAR equations
- Publication
Mathematical Proceedings of the Cambridge Philosophical Society, 2014, Vol 156, Issue 1, p167
- ISSN
0305-0041
- Publication type
Article
- DOI
10.1017/S0305004113000480