Multiple solutions for the logarithmic fractional Kirchhoff equation with critical or supercritical nonlinearity

Abstract

In this paper, we study the following logarithmic fractional Kirchhoff equation: ( a + b ∫ R 3 | ( − Δ ) s / 2 u | 2 d x ) ( − Δ ) s u + V ( x ) u = | u | p − 2 u log ⁡ u 2 + λ | u | q − 2 u , in R 3 , where a, b>0, ( − Δ ) s is the fractional Laplace operator with q ≥ 2 s ∗ = 6 3 − 2 s and V satisfies suitable conditions. Since the presence of supercritical exponents leads to a lack of compactness, we introduce a truncation function to reduce the exponents, allowing compactness to be recovered. We show that the above equation has a ground state solution and a sign-changing solution via the constraint variation method, the quantitative deformation lemma and Moser's iterative method.