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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

-Δu(x)+V(x)u(x)=K(x)f(u),x N ,uH 1 ( N ),u(x)>0,N3,(1)

where the potential V(x) may vanish at infinity, f(s) is asymptotically linear at infinity, that is, f(s)O(s) as s+. For this kind of potential, it seems difficult to find solutions in H 1 ( N ), i.e. bound states of (1). If f(s)=s p and p(σ,(N+2)/(N-2)) with σ1, 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 ( N ) and (1) has no ground states if p is out of the above range. We are interested in what happens if f(s) is asymptotically linear. Under appropriate assumptions on K, we prove that (1) has a bound state and a ground state.

35Q55NLS-like (nonlinear Schrödinger) equations
35J20Second order elliptic equations, variational methods
35J60Nonlinear elliptic equations
81Q05Closed and approximate solutions to quantum-mechanical equations
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