Ground states and concentration phenomena for the fractional Schrödinger equation

Abstract

Abstract. We consider here solutions of the nonlinear fractional Schrödinger equationWe show that concentration points must be critical points for V . We also prove that, if the potential V is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point as ε tends to zero. In addition, if the potential V is radial and radially decreasing, then the minimizer is unique provided ε is small.

Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation

“…Set φ ε = ϕ − v j ε . It follows from (57) and (58) that Arguing in a similar way as [11], we get φ ε (x) ≥ 0 for all |x| ≥ R. Hence, we obtain…”

“…These solutions concentrate at the global minimum point of V as ε tends to zero, was showed by [11] for f (u) = u p , 1 < p < N +2s N −2s , that is the equation ε 2s (−∆) s u + V (x)u − u p = 0,…”

“…In the recent years, nonlinear Schrödinger equations involving the fractional Laplacian have been studied extensively by many authors. See for example [9,10,11,12,17] and the references therein. In [17], Secchi used variational method to study the equation…”

Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent

“…We also mention the papers that Chang and Wang [3], Dipierro et al [8], Secchi [20], and Teng [26] studied the subcritical growth case using the variational approach, respectively. In [7] and [12], the concentration phenomenon was considered. In the critical case, in [4], Chen, Li and Ou studied the existence, uniquness and symmetry properties for critical fractional Schrödinger equations.…”

Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential

“…Then there existsΛ 0 > 0 such that for every λ >Λ 0 , equation (1.1) has at least two non-trivial non-negative solutions u + λ and u − λ . Recently, concentration phenomena for the fractional Schrödinger equation has been attracted many attentions, see for example [9,10,16,27] and references therein. When V λ = λV + , Torres [27] consider the following problem…”