Asymptotic Analysis

In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension d ⩾ 1 , in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form ( 1 + | v | 2 ) − β 2 , in the range β ∈ ] d , d + 4 [ . The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation (Nonlinearity 28 (2015) 545). The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker–Planck equation, for the correct density ρ : = ∫ R d f d v , with a fractional Laplacian κ ( − Δ ) β − d + 2 6 and a positive diffusion coefficient κ.

In this paper, a class of ( p , q ) -Laplacian equations with critical growth is taken into consideration: − Δ p u − Δ q u + ( | u | p − 2 + | u | q − 2 ) u + λ ϕ | u | q − 2 u = μ g ( u ) + | u | q ∗ − 2 u , x ∈ R 3 , − Δ ϕ = | u | q , x ∈ R 3 , where Δ ξ u = div ( | ∇ u | ξ − 2 ∇ u ) is the ξ-Laplacian operator ( ξ = p , q ) , 3 2 < p < q < 3 , λ and μ are positive parameters, q ∗ = 3 q / ( 3 − q ) is the Sobolev critical exponent. We use a primary technique of constrained minimization to determine the existence, energy estimate and convergence property of nodal (that is, sign-changing) solutions under appropriate conditions on g, and thus generalize the existing results.