Existence and Non-existence of Extremals for Critical Adams Inequality in any Even Dimension

Recently, the authors of the current paper established the existence and non-existence of extremals of the second order critical Adams inequalities in R⁴ (see Adv Math 368:107143 (2020)). One of the crucial elements used in [8] is the accurate upper-bound of the following concentration limit as β_k→ 32 π²: limsupk→+∞∫R4(exp(βkuk2)-1-αuk2)dx≤π26exp(53+32π2A0),where u_k is the maximizing sequence corresponding to the critical Adams inequality on R⁴. This was obtained through finding the precise expression of the optimal bi-harmonic truncation which generates the smallest energy, where A is the value at 0 of the trace of the Green function’s regular part for the operator Δ ²+ 1. However, such a precise expression for the optimal poly-harmonic truncation ukδ(x) in any even dimension space R^2m is still unknown. See formulas (3.24) and (3.25). Furthermore, it is also a difficult task to obtain the accurate estimate for the Dirichlet energy of uk(x)-ukδ(x) on small ball B_δ(x_k) : ∫Bδ(xk)|Δm2(uk(x)-ukδ(x))|2dxin the case of the Adams inequality of higher order derivative m with m> 2 (see 3.29). This can be computed explicitly by direct computation in the case m= 1 or m= 2 , but much more difficult to do for m> 2. These obstructions will be overcome in the present paper by using some novel methods. As a result, we will establish the existence and non-existence of extremal functions for the m-th order Adams inequality in any even dimensional space R^2m which was left open for m> 2 in [8]. Our proofs are partly based on the sharp Fourier rearrangement inequality, the blow-up analysis of the Euler-Lagrange equations for the critical Adams functional, Boggio’s formula for the Green function of Δ ^m in the ball B_R and careful and delicate analysis of the optimal poly-harmonic truncations.