Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal <i>L</i>2-critical or <i>L</i>2-supercritical perturbation
Tao Yang
Abstract
In this paper, we study the existence and asymptotic properties of solutions to the fractional Schrödinger equation (−Δ)σu=λu+|u|q−2u+μIα*|u|p|u|p−2u under the normalized constraint ∫RNu2=a2, where N ≥ 2, σ ∈ (0, 1), α ∈ (0, N), q∈(2+4σN,2NN−2σ], p∈[1+2σ+αN,N+αN−2σ), a, μ &gt; 0, Iα(x) = |x|α−N, and λ∈R appears as a Lagrange multiplier. By using a refined version of the min-max principle, we show that the above problem admits a mountain pass type solution ûμ for some λ̂&lt;0 under suitable assumptions on the related parameters. In particular, we can prove that ûμ is a ground state if p≤q2+αN. Furthermore, we give some asymptotic properties of the solutions. We mainly extend the results in the work of Bhattarai [J. Differ. Equations 263, 3197–3229 (2017)] and Feng et al. [J. Math. Phys. 60, 1–12(2019)] concerning the above problem from the L2-subcritical setting to L2-critical and L2-supercritical settings with respect to p, involving the Sobolev critical case q=2NN−2σ especially.