Ambrosetti, A.; Badiale, M.; Cingolani, S. Semiclassical states of nonlinear Schrödinger equations. (English) Zbl 0896.35042 Arch. Ration. Mech. Anal. 140, No. 3, 285-300 (1997). The authors investigate the solutions \(u\) of the equation \[ -h^2\Delta u+\lambda u+ V(x)u= | u|^{p-1}u\quad\text{in }\mathbb{R}^N, \] where \(\lambda>0\), \(p<(N+ 2)/(N- 2)\), and \(V\) admits a critical point at \(0\). Using variational methods, they prove the existence of a solution which is semiclassically localized at \(0\), in the sense that at any fixed \(x\neq 0\), the solution decays exponentially as \(R\to 0_+\). Reviewer: André Martinez (Bologna) Cited in 3 ReviewsCited in 280 Documents MSC: 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 35B40 Asymptotic behavior of solutions to PDEs Keywords:semiclassically localized solution PDF BibTeX XML Cite \textit{A. Ambrosetti} et al., Arch. Ration. Mech. Anal. 140, No. 3, 285--300 (1997; Zbl 0896.35042) Full Text: DOI OpenURL