On some weighted fourth-order equations

This paper deals with the following radial Caffarelli-Kohn-Nirenberg-type inequality,[GRAPHICS]where N >= 3, 2 < alpha < N, l = 4(alpha-2)(N-2)/N-alpha - alpha and p(alpha)* = 2(N+l)/N-4+alpha. Then we consider the related Euler-Lagrange equation:[GRAPHICS]For alpha not equal 0 or l not equal 0, it is known the solutions of above equation are invariant for dilations lambda(N-4+alpha/2) u(lambda x) but not for translations. However we show that if alpha is an even integer, there exist new solutions to the linearized problem, related to the radial solution that "replace" the ones due to the translations invariance. As an application, we turn back to investigate the remainder terms of the above inequality. As well as some fourth-order Liouville-type equations with singular data. (c) 2023 Elsevier Inc. All rights reserved.