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Semi-classical bound states of Schrödinger equations

Published online by Cambridge University Press:  21 October 2013

M. SCHECHTER
Affiliation:
Department of Mathematics, University of California Irvine, CA 92697-3875, U.S.A. e-mail: mschecht@math.uci.edu
W. ZOU
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China. e-mail: wzou@math.tsinghua.edu.cn
Corresponding

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2013 

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