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

##### 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)
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##### References:
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