## Abstract

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

$\left\{\begin{array}{cc}-\mathrm{\Delta}u-\gamma \left(u\u2215|x{|}^{2}\right)-\mathit{\epsilon}u=|u{|}^{\frac{4}{N-2}}u\phantom{\rule{1em}{0ex}}\hfill & \text{in}\mathrm{\Omega},\hfill \\ u=0\phantom{\rule{1em}{0ex}}\hfill & \text{on}\partial \mathrm{\Omega},\hfill \end{array}\right.$

when $\mathit{\epsilon}>0$ is small, , and where $\mathrm{\Omega}\subset {\mathbb{R}}^{N}$, $N\ge 3$, is a smooth bounded domain with $0\in \mathrm{\Omega}$. We show that there exists a sequence ${\left({\gamma}_{j}\right)}_{j=1}^{\infty}$ in $\left(-\infty ,0\right]$ with ${lim}_{j\to \infty}{\gamma}_{j}=-\infty $ such that, if $\gamma \ne {\gamma}_{j}$ for any $j$ and $\gamma \le \frac{{\left(N-2\right)}^{2}}{4}-1$, then the above equation has for $\mathit{\epsilon}$ small, a positive — in general nonminimizing — solution that develops a bubble at the origin. If moreover , then for any integer $k\ge 2$, the equation has for small enough $\mathit{\epsilon}$ 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 $\gamma \ne {\gamma}_{j}$ is not necessary. Indeed, it is known that, if $\gamma >\frac{{\left(N-2\right)}^{2}}{4}-1$ and $\mathrm{\Omega}$ is a ball $B$, then there is no radial positive solution for $\mathit{\epsilon}>0$ small. We complete the picture here by showing that, if $\gamma \ge \frac{{\left(N-2\right)}^{2}}{4}-4$, then the above problem has no radial sign-changing solutions for $\mathit{\epsilon}>0$ small. These results recover and improve what is already known in the nonsingular case, i.e., when $\gamma =0$.

## Citation

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

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