Sign-changing solutions for a fractional Choquard equation with power nonlinearity. (August 2022) - Zhao, Shunneng; Yu, Yuanyang
Abstract: In this paper, we study the following fractional Choquard equation (FC) ( − Δ ) s u + V ( x ) u = ( I α ∗ | u | p ) | u | p − 2 u, in R N, where s ∈ ( 0, 1 ), N ≥ 3, V ( x ) is continuous potential function, I α : R N → R is the Riesz potential of order α defined by I α ( x ) = | x | α − N for every x ∈ R N ∖ 0 , α ∈ ( 0, N ), ∗ denotes the convolution operator, 2 < p < N + α N − 2 s = 2 α, s ∗, 2 α, s ∗ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and fractional Laplace operator and the operator ( − Δ ) s stands for the fractional Laplacian of order s . Combining constraint variational method, quantitative deformation lemma and the Brouwer degree theory, we prove that (FC) possesses one least energy sign-changing solution u 0 . Moreover, we show that the energy of u 0 is strictly larger than 2 p − 2 p − 1 times and strictly less than four times the ground state energy.