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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 $$-\varepsilon^2\Delta v+V(x)v=K(x)v^p,\quad x\in \bbfR^N,$$ 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 $W^{1,2}(\bbfR^N)$ 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 ${\cal A}=V^\theta K^{-2/(p-1)}$, where $\theta= (p+1)/(p-1)-1/2$.

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)
Full Text: DOI Link
[1] Ambrosetti, A., Badiale, M.: Homoclinics: Poincaré-Melnikov type results via a variational approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 , 233-252 (1998) · Zbl 1004.37043 · doi:10.1016/S0294-1449(97)89300-6 · numdam:AIHPC_1998__15_2_233_0 · eudml:78437
[2] Ambrosetti, A., Badiale, M.: Variational perturbative methods and bifurcation of bound states from the essential spectrum. Proc. Roy. Soc. Edinburgh Sect. A 128 , 1131-1161 (1998) · Zbl 0928.34029 · doi:10.1017/S0308210500027268
[3] Ambrosetti, A., Badiale, M., Cingolani, S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Rat. Mech. Anal. 140 , 285-300 (1997) · Zbl 0896.35042 · doi:10.1007/s002050050067
[4] Ambrosetti, A., Garcia Azorero, J., Peral, I.: Remarks on a class of semilinear elliptic equa- tions on n R , via perturbation methods. Adv. Nonlinear Stud. 1 , 1-13 (2001) · Zbl 1001.35038