Revista Matemática Iberoamericana

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Volume 24, Issue 1, 2008, pp. 297–351
DOI: 10.4171/RMI/537

Published online: 2008-04-30

Bound state solutions for a class of nonlinear Schrödinger equations

Denis Bonheure[1] and Jean Van Schaftingen[2]

(1) Université libre de Bruxelles, Belgium
(2) Université Catholique de Louvain, Belgium

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form $$ -\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{R}^N, $ where $V, K$ are positive continuous functions and $p > 1$ is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential $V$ is allowed to vanish at infinity and the competing function $K$ does not have to be bounded. In the \emph{semi-classical limit}, i.e. for $\varepsilon\sim 0$, we prove the existence of bound state solutions localized around local minimum points of the auxiliary function $\mathcal{A} = V^\theta K^{-\frac{2}{p-1}}$, where $\theta=(p+1)/(p-1)-N/2$. A special attention is devoted to the qualitative properties of these solutions as $\varepsilon$ goes to zero.

Keywords: Nonlinear Schrödinger equation, semi-classical states, concentration, vanishing potentials, unbounded competition functions

Bonheure Denis, Van Schaftingen Jean: Bound state solutions for a class of nonlinear Schrödinger equations. Rev. Mat. Iberoam. 24 (2008), 297-351. doi: 10.4171/RMI/537