Semiclassical states of nonlinear Schrödinger equations. (English) Zbl 0896.35042

Ambrosetti, A.; Badiale, M.; Cingolani, S.

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)

MSC:

35J60	Nonlinear elliptic equations
35J20	Variational methods for second-order elliptic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

semiclassically localized solution