[HTML][HTML] Concentrating standing waves for the fractional nonlinear Schrödinger equation

J Dávila, M Del Pino, J Wei - Journal of Differential Equations, 2014 - Elsevier
We consider the semilinear equation ε 2 s (− Δ) s u+ V (x) u− up= 0, u> 0, u∈ H 2 s (RN)
where 0< s< 1, 1< p< N+ 2 s N− 2 s, V (x) is a sufficiently smooth potential with inf RV (x)> 0,
and ε> 0 is a small number. Letting w λ be the radial ground state of (− Δ) sw λ+ λ w λ− w λ
p= 0 in H 2 s (RN), we build solutions of the form u ε (x)∼∑ i= 1 kw λ i ((x− ξ i ε)/ε), where λ i=
V (ξ i ε) and the ξ i ε approach suitable critical points of V. Via a Lyapunov–Schmidt
variational reduction, we recover various existence results already known for the case s= 1 …

Concentrating standing waves for the fractional nonlinear Schrodinger equation

J Dávila Bonczos, M Pino Manresa, J Wei - 2014 - repositorio.uchile.cl
We consider the semilinear equation epsilon (2s)(-Delta)(s) u+ V (x) uu (p)= 0, u> 0, u is an
element of H-2s (RN) where 0< s< 1, 1< p< N+ 2s/N-2s, V (x) is a sufficiently smooth
potential with inf (R) V (x)> 0, and epsilon> 0 is a small number. Letting w (lambda) be the
radial ground state of (-Delta)(s) w (lambda)+ lambda w (lambda)-w (lambda)(p)= 0 in H-2s
(RN), we build solutions of the form u epsilon (x) similar to (k) Sigma (i= 1) w lambda (i)((x-xi
(epsilon)(i))/epsilon), where lambda (i)= V (xi (epsilon)(i)) and the xi (epsilon)(i) approach …
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