Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity
Summary: The authors deal with a class on nonlinear Schrödinger equations \[ -\varepsilon^2\Delta v+V(x)v=K(x)v^p,\quad x\in \mathbb{R}^N, \] 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 \(W^{1,2}(\mathbb{R}^N)\) 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 \({\mathcal A}=V^\theta K^{-2/(p-1)}\), where \(\theta= (p+1)/(p-1)-1/2\).
scientific article; zbMATH DE number 2148279