2019/02/28 by Nguyen, Van Hoang
#26D10 #26D15 #35A23 #46E35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1902.10970
In this paper, we establish the following Leray--Adams type inequality on a bounded domain Ω in \mathbb R4 containing the origin, supu∈ C0^∞(Ω), I4[u,Ω,R] ≤ 1 ∫Ωexp(c( \frac|u|E2β(\frac|x|R))2) dx ≤ C |Ω| for some constants c >0 and C >0, where β≥ 1, R ≥ supx∈ Ω |x|, I4[u,Ω,R]:= ∫Ω|Δu|2 dx - ∫Ω\frac|u|2|x|4 E12(\frac|x|R) dx, and E1(t) = 1-ln t, E2(t) = ln (eE1(t)) for t ∈ (0,1]. This extends the Leray--Trudinger inequality recently established by Psaradakis and Spector \citePS2015 and Mallick and Tintarev \citeMT2018 to the case of Laplacian operator. In the higher dimensions or higher order derivatives, we prove the Leray--Adams type inequality for radial function on the ball Br (with center at origin and radius r >0) in \mathbb Rn.