Sign-changing solutions for critical equations with Hardy potential

Abstract

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

when is small, , and where $\mathrm{\Omega }\subset {ℝ}^N$, $N\ge 3$, 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 $𝜖$ 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 $𝜖$ 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 and $\mathrm{\Omega }$ is a ball $B$, then there is no radial positive solution for 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 small. These results recover and improve what is already known in the nonsingular case, i.e., when $\gamma =0$.