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Asymptotically linear Schrödinger equation with potential vanishing at infinity. (English) Zbl 1188.35181

Summary: We are concerned with the existence of bound states and ground states of the following nonlinear Schrödinger equation

$-{\Delta }u\left(x\right)+V\left(x\right)u\left(x\right)=K\left(x\right)f\left(u\right),\phantom{\rule{1.em}{0ex}}x\in {ℝ}^{N},\phantom{\rule{2.em}{0ex}}u\in {H}^{1}\left({ℝ}^{N}\right),\phantom{\rule{1.em}{0ex}}u\left(x\right)>0,\phantom{\rule{1.em}{0ex}}N\ge 3,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where the potential $V\left(x\right)$ may vanish at infinity, $f\left(s\right)$ is asymptotically linear at infinity, that is, $f\left(s\right)\sim O\left(s\right)$ as $s\to +\infty$. For this kind of potential, it seems difficult to find solutions in ${H}^{1}\left({ℝ}^{N}\right)$, i.e. bound states of (1). If $f\left(s\right)={s}^{p}$ and $p\in \left(\sigma ,\left(N+2\right)/\left(N-2\right)\right)$ with $\sigma \ge 1$, A. Ambrosetti, V. Felli and A. Malchiodi [J. Eur. Math. Soc. (JEMS) 7, No. 1, 117–144 (2005; Zbl 1064.35175)] showed that (1) has a solution ${H}^{1}\left({ℝ}^{N}\right)$ and (1) has no ground states if $p$ is out of the above range. We are interested in what happens if $f\left(s\right)$ is asymptotically linear. Under appropriate assumptions on $K$, we prove that (1) has a bound state and a ground state.

##### MSC:
 35Q55 NLS-like (nonlinear Schrödinger) equations 35J20 Second order elliptic equations, variational methods 35J60 Nonlinear elliptic equations 81Q05 Closed and approximate solutions to quantum-mechanical equations
##### References:
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