## Revista Matemática Iberoamericana

### Bound state solutions for a class of nonlinear Schrödinger equations

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schr�dinger 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. #### Article information Source Rev. Mat. Iberoamericana Volume 24, Number 1 (2008), 297-351. Dates First available in Project Euclid: 16 July 2008 Permanent link to this document https://projecteuclid.org/euclid.rmi/1216247103 Mathematical Reviews number (MathSciNet) MR2435974 Zentralblatt MATH identifier 1156.35084 #### Citation Bonheure , Denis; Van Schaftingen , Jean. Bound state solutions for a class of nonlinear Schrödinger equations. Rev. Mat. Iberoamericana 24 (2008), no. 1, 297--351.https://projecteuclid.org/euclid.rmi/1216247103 #### References • Ambrosetti, A., Badiale, M. and Cingolani, S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 140 (1997), 285-300. • Ambrosetti, A., Felli, V. and Malchiodi, A.: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. (JEMS) 7 (2005), 117-144. • Ambrosetti, A. and Malchiodi, A.: Concentration phenomena for nonlinear Schrödinger equation: recent results and new perspectives. 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