Vol. 14, No. 2, 2021

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Sign-changing solutions for critical equations with Hardy potential

Pierpaolo Esposito, Nassif Ghoussoub, Angela Pistoia and Giusi Vaira

Vol. 14 (2021), No. 2, 533–566

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

Δu γ(u|x|2) 𝜖u = |u| 4 N2 u in Ω, u = 0  on Ω,

when 𝜖 > 0 is small, γ < (N2)2 4 , and where Ω N , N 3, is a smooth bounded domain with 0 Ω. We show that there exists a sequence (γj)j=1 in (,0] with limjγj = such that, if γγj for any j and γ (N2)2 4 1, 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)2 4 4, then for any integer k 2, 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)2 4 1 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)2 4 4, 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.

critical problem, Hardy potential, linear perturbation, blow-up point
Mathematical Subject Classification 2010
Primary: 35A15, 35J20, 35J47, 35B33, 35B44, 35J75, 35J91
Received: 2 October 2018
Revised: 20 May 2019
Accepted: 25 October 2019
Published: 20 March 2021
Pierpaolo Esposito
Dipartimento di Matematica e Fisica
Università degli Studi Roma Tre
Nassif Ghoussoub
Department of Mathematics
University of British Columbia
Vancouver, BC
Angela Pistoia
Dipartimento di Metodi e Modelli Matematici
Università di Roma “La Sapienza”
Giusi Vaira
Dipartimento di Matematica e Fisica
Sapienza Università degli Studi della Campania “Luigi Vanvitelli”