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
Abstract

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

when $𝜖>0$ is small, $\gamma <\frac{{\left(N-2\right)}^{2}}{4}$, and where $\Omega \subset {ℝ}^{N}$, $N\ge 3$, is a smooth bounded domain with $0\in \Omega$. We show that there exists a sequence ${\left({\gamma }_{j}\right)}_{j=1}^{\infty }$ in $\left(-\infty ,0\right]$ with $\underset{j\to \infty }{lim}{\gamma }_{j}=-\infty$ 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 $𝜖$ small, a positive — in general nonminimizing — solution that develops a bubble at the origin. If moreover $\gamma <\frac{{\left(N-2\right)}^{2}}{4}-4$, then for any integer $k\ge 2$, 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 $\gamma \ne {\gamma }_{j}$ is not necessary. Indeed, it is known that, if $\gamma >\frac{{\left(N-2\right)}^{2}}{4}-1$ and $\Omega$ is a ball $B$, then there is no radial positive solution for $𝜖>0$ 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 $𝜖>0$ small. These results recover and improve what is already known in the nonsingular case, i.e., when $\gamma =0$.

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