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.

On some weighted fourth-order equations / Deng, Shengbing; Grossi, Massimo; Tian, Xingliang. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 364:(2023), pp. 612-634. [10.1016/j.jde.2023.03.054]

On some weighted fourth-order equations

Massimo Grossi
;
2023

Abstract

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.
2023
Caffarelli-Kohn-Nirenberg inequalities; Hardy-Henon equation; Non-degeneracy; Remainder terms
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On some weighted fourth-order equations / Deng, Shengbing; Grossi, Massimo; Tian, Xingliang. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 364:(2023), pp. 612-634. [10.1016/j.jde.2023.03.054]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1686615
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