Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity
Copyright policy )Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity
We deal with a class on nonlinear Schrödinger equations (NLS) 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 of {\mathcal A}=V^{\theta}K^{-2/(p-1)} , where \theta= (p+1)/(p-1)-1/2 .
Variational methods for second-order elliptic equations, nonlinear Schrödinger equations, Applied Mathematics, General Mathematics, NLS equations (nonlinear Schrödinger equations), Nonlinear Schrödinger equations; weighted Sobolev spaces, existence of ground states, critical point, Nonlinear Schrödinger equations, ground states, Existence of generalized solutions of PDE, Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces, Nonlinear elliptic equations, bound states, weighted Sobolev spaces, Variational methods involving nonlinear operators, standing wave, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, Nonlinear Schrödinger equations; Weighted Sobolev spaces;, Singular perturbations in context of PDEs
Variational methods for second-order elliptic equations, nonlinear Schrödinger equations, Applied Mathematics, General Mathematics, NLS equations (nonlinear Schrödinger equations), Nonlinear Schrödinger equations; weighted Sobolev spaces, existence of ground states, critical point, Nonlinear Schrödinger equations, ground states, Existence of generalized solutions of PDE, Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces, Nonlinear elliptic equations, bound states, weighted Sobolev spaces, Variational methods involving nonlinear operators, standing wave, Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, Nonlinear Schrödinger equations; Weighted Sobolev spaces;, Singular perturbations in context of PDEs
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