We consider the following perturbed critical Dirichlet problem involving the Hardy-Schr"odinger operator on a smooth bounded domain $Omega subset mathbbR^N$, $Ngeq 3$, with $0 in Omega$: $$ left eginarrayll-Delta u-gamma racu|x|^2-epsilon u=|u|^rac4N-2u &hboxin Omega u=0 & hboxon partial Omega, endarray ight. $$ when $epsilon>0$ is small and $gamma< (N-2)^2over4$. Setting $ gamma_j= rac(N-2)^24left(1-racj(N-2+j)N-1 ight)in(-infty,0]$ for $j in mathbbN,$ we show that if $gammaleq rac(N-2)^24-1$ and $gamma eq gamma_j$ for any $j$, then for small $epsilon$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $gamma rac(N-2)^24-1$ and $Omega$ is a ball $B$, then there is no radial positive solution for $epsilon>0$ small. We complete the picture here by showing that, if $gammageq rac(N-2)^24-4$, then the above problem has no radial sign-changing solutions for $epsilon>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $gamma=0$.

Esposito, P., Ghoussoub, N., Pistoia, A., Vaira, G. (2021). Sign-Changing Solutions for Critical Equations with Hardy Potential. ANALYSIS & PDE, 14(2), 533-566 [10.2140/apde.2021.14.533].

Sign-Changing Solutions for Critical Equations with Hardy Potential

Pierpaolo Esposito
;
2021-01-01

Abstract

We consider the following perturbed critical Dirichlet problem involving the Hardy-Schr"odinger operator on a smooth bounded domain $Omega subset mathbbR^N$, $Ngeq 3$, with $0 in Omega$: $$ left eginarrayll-Delta u-gamma racu|x|^2-epsilon u=|u|^rac4N-2u &hboxin Omega u=0 & hboxon partial Omega, endarray ight. $$ when $epsilon>0$ is small and $gamma< (N-2)^2over4$. Setting $ gamma_j= rac(N-2)^24left(1-racj(N-2+j)N-1 ight)in(-infty,0]$ for $j in mathbbN,$ we show that if $gammaleq rac(N-2)^24-1$ and $gamma eq gamma_j$ for any $j$, then for small $epsilon$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $gamma rac(N-2)^24-1$ and $Omega$ is a ball $B$, then there is no radial positive solution for $epsilon>0$ small. We complete the picture here by showing that, if $gammageq rac(N-2)^24-4$, then the above problem has no radial sign-changing solutions for $epsilon>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $gamma=0$.
Esposito, P., Ghoussoub, N., Pistoia, A., Vaira, G. (2021). Sign-Changing Solutions for Critical Equations with Hardy Potential. ANALYSIS & PDE, 14(2), 533-566 [10.2140/apde.2021.14.533].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/340925
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