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Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. (English) Zbl 1064.35175

Summary: The authors deal with a class on nonlinear Schrödinger equations

$-{\epsilon }^{2}{\Delta }v+V\left(x\right)v=K\left(x\right){v}^{p},\phantom{\rule{1.em}{0ex}}x\in {ℝ}^{N},$

with potentials $V\left(x\right)\sim {|x|}^{-\alpha }$, $0<\alpha <2$, and $K\left(x\right)\sim {|x|}^{-\beta }$, $\beta >0$. Working in weighted Sobolev spaces, the existence of ground states ${v}_{\epsilon }$ belonging to ${W}^{1,2}\left({ℝ}^{N}\right)$ is proved under the assumption that $\sigma for some $\sigma ={\sigma }_{N,\alpha ,\beta }$. Furthermore, it is shown that ${v}_{\epsilon }$ are spikes concentrating at a minimum point of $𝒜={V}^{\theta }{K}^{-2/\left(p-1\right)}$, where $\theta =\left(p+1\right)/\left(p-1\right)-1/2$.

##### MSC:
 35Q55 NLS-like (nonlinear Schrödinger) equations 81Q05 Closed and approximate solutions to quantum-mechanical equations 35J20 Second order elliptic equations, variational methods 35J60 Nonlinear elliptic equations 35B25 Singular perturbations (PDE) 47J30 Variational methods (nonlinear operator equations)