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

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 \(R\to 0_+\).

MSC:

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