The concentration of solutions to a fractional Schrödinger equation

Existence and mass concentration of pseudo-relativistic Hartree equation

“…The limit behavior of the minimizer u a k as a k → a * is also investigated. Similar problems were considered in [4] for the fractional Laplacian (−∆) s and in [9] for Choquard equation.…”

Existence, non-existence and blow-up behaviour of minimizers for the mass-critical fractional non-linear Schrödinger equations with periodic potentials

“…In this paper, we give an alternative approach based on the compactness of optimizing sequence for the fractional Gagliardo-Nirenberg inequality. Our argument is simpler and more direct than the ones in [11,22]. We finally point out that in contrast of the classical Schrödinger equation s = 1 in which the ground state decays exponentially at infinity, the ground state for fractional Schrödinger equation 0 < s < 1 decays only polynomially at infinity.…”

“…In the case 0 < s < 1, the existence, non-existence and blow-up behaviour of minimizers for I(a) has been considered in several works. In [22], He-Long proved the existence and non-existence of minimizers for I(a) with trapping potentials (1.6) and bounded potentials satisfying (1.9). They also studied the blow-up behaviour of minimizers for I(a) as a tends to a critical value in the mass-critical case α = 4s/d.…”

“…The proof of theorem 1.5 is based on energy estimates (see lemma 4.1). In [11,22], these energy estimates were obtained by analysing the corresponding Euler-Lagrange equation related to the minimizers. In this paper, we give an alternative approach based on the compactness of optimizing sequence for the fractional Gagliardo-Nirenberg inequality.…”

Asymptotic behavior of normalized ground states for the fractional Schrödinger equation with combined L2‐critical and L2‐subcritical nonlinearities

“…After this work, problems of this type have been extensively studied; see previous studies. [22][23][24][25][26][27][28][29][30][31][32][33][34][35] Motivated by the above aforementioned papers, we study the asymptotic behavior of minimizers for (7) as c ↗ c * . We can obtain the following results.…”