Sign-changing solutions for critical equations with Hardy potential

when $𝜖 > 0$ is small, $γ < \frac{{(N - 2)}^{2}}{4}$ , and where $Ω \subset ℝ^{N}$ , $N \geq 3$ , is a smooth bounded domain with $0 \in Ω$ . We show that there exists a sequence ${(γ_{j})}_{j = 1}^{\infty}$ in $(- \infty, 0]$ with $lim_{j \to \infty} γ_{j} = - \infty$ such that, if $γ \neq γ_{j}$ for any $j$ and $γ \leq \frac{{(N - 2)}^{2}}{4} - 1$ , then the above equation has for $𝜖$ small, a positive — in general nonminimizing — solution that develops a bubble at the origin. If moreover $γ < \frac{{(N - 2)}^{2}}{4} - 4$ , then for any integer $k \geq 2$ , 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 $γ \neq γ_{j}$ is not necessary. Indeed, it is known that, if $γ > \frac{{(N - 2)}^{2}}{4} - 1$ 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 $γ \geq \frac{{(N - 2)}^{2}}{4} - 4$ , 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 $γ = 0$ .

Keywords

critical problem, Hardy potential, linear perturbation, blow-up point

Mathematical Subject Classification 2010

Primary: 35A15, 35J20, 35J47, 35B33, 35B44, 35J75, 35J91

Milestones

Received: 2 October 2018

Revised: 20 May 2019

Accepted: 25 October 2019

Published: 20 March 2021

Authors

Pierpaolo Esposito
Dipartimento di Matematica e Fisica Università degli Studi Roma Tre Roma Italy

Nassif Ghoussoub
Department of Mathematics University of British Columbia Vancouver, BC Canada

Angela Pistoia
Dipartimento di Metodi e Modelli Matematici Università di Roma “La Sapienza” Roma Italy

Giusi Vaira
Dipartimento di Matematica e Fisica Sapienza Università degli Studi della Campania “Luigi Vanvitelli” Caserta Italy

Vol. 14, No. 2, 2021

Pierpaolo Esposito, Nassif Ghoussoub, Angela Pistoia and Giusi Vaira

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors