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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

-ε 2 Δv+V(x)v=K(x)v p ,x N ,

with potentials V(x)|x| -α , 0<α<2, and K(x)|x| -β , β>0. Working in weighted Sobolev spaces, the existence of ground states v ε belonging to W 1,2 ( N ) is proved under the assumption that σ<p<(N+2)/(N-2) for some σ=σ N,α,β . Furthermore, it is shown that v ε are spikes concentrating at a minimum point of 𝒜=V θ K -2/(p-1) , where θ=(p+1)/(p-1)-1/2.


MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
81Q05Closed and approximate solutions to quantum-mechanical equations
35J20Second order elliptic equations, variational methods
35J60Nonlinear elliptic equations
35B25Singular perturbations (PDE)
47J30Variational methods (nonlinear operator equations)