Sign-Changing Solutions for Critical Equations with Hardy Potential

Abstract

We consider the following perturbed critical Dirichlet problem involving the Hardy-Schrödinger operator on a smooth bounded domain $Ω\subset \mathbb{R}^N$, $N\geq 3$, with $0 \in Ω$: $$ \left\{ \begin{array}{ll}-Δu-γ\frac{u}{|x|^2}-εu=|u|^{\frac{4}{N-2}}u &\hbox{in }Ωu=0 & \hbox{on }\partial Ω, \end{array}\right. $$ when $ε>0$ is small and $γ< {(N-2)^2\over4}$. Setting $ γ_j= \frac{(N-2)^2}{4}\left(1-\frac{j(N-2+j)}{N-1}\right)\in(-\infty,0]$ for $j \in \mathbb{N},$ we show that if $γ\leq \frac{(N-2)^2}{4}-1$ and $γ\neq γ_j$ for any $j$, then for small $ε$, the above equation has a positive --non variational-- 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 that $γ\neq γ_j$ 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 known in the non-singular case, i.e., when $γ=0$.