## 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

### Keywords:

semiclassically localized solution
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