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On the shape of the least-energy solutions to some singularly perturbed mixed problems
A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials
| 1. | Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O.Box 71010, Wuhan 430071, China, China |
$-\Delta u + V(|x|)u =Q(|x|)|u|^{p-2}u, x\in R^N, $
$u(x)\rightarrow 0$ as $|x|\rightarrow+\infty,$
where $N\geq 3$, $p\in(2,+\infty)$, $V(x)$ and $Q(x)$ are continuous functions which vanishes at infinity and may change sign. As a special case, our result shows that the solutions obtained by Su-Wang-Willem in [11, Theorem 3] must decay precisely like $|x|^{-(N-2)}$ as $|x|\rightarrow+\infty$ if $V(|x|)$ decays faster than $|x|^{-2}$ at infinity.
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