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2021 Sign-changing solutions for critical equations with Hardy potential
Pierpaolo Esposito, Nassif Ghoussoub, Angela Pistoia, Giusi Vaira
Anal. PDE 14(2): 533-566 (2021). DOI: 10.2140/apde.2021.14.533


We consider the following perturbed critical Dirichlet problem involving the Hardy–Schrödinger operator:

Δuγ(u|x|2)𝜖u=|u|4N2u in Ω,u=0 on Ω,

when 𝜖>0 is small, γ<(N2)24, and where ΩN, N3, is a smooth bounded domain with 0Ω. We show that there exists a sequence (γj)j=1 in (,0] with limjγj= such that, if γγj for any j and γ(N2)241, then the above equation has for 𝜖 small, a positive — in general nonminimizing — solution that develops a bubble at the origin. If moreover γ<(N2)244, then for any integer k2, the equation has for small enough 𝜖 a sign-changing solution that develops into a superposition of k bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition γγj is not necessary. Indeed, it is known that, if γ>(N2)241 and Ω is a ball B, then there is no radial positive solution for 𝜖>0 small. We complete the picture here by showing that, if γ(N2)244, then the above problem has no radial sign-changing solutions for 𝜖>0 small. These results recover and improve what is already known in the nonsingular case, i.e., when γ=0.


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Pierpaolo Esposito. Nassif Ghoussoub. Angela Pistoia. Giusi Vaira. "Sign-changing solutions for critical equations with Hardy potential." Anal. PDE 14 (2) 533 - 566, 2021.


Received: 2 October 2018; Revised: 20 May 2019; Accepted: 25 October 2019; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/apde.2021.14.533


Rights: Copyright © 2021 Mathematical Sciences Publishers


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Vol.14 • No. 2 • 2021
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