We consider the following perturbed critical Dirichlet problem involving the Hardy–Schrödinger operator:
when is small, , and where , , is a smooth bounded domain with . We show that there exists a sequence in with such that, if for any and , then the above equation has for small, a positive — in general nonminimizing — solution that develops a bubble at the origin. If moreover , then for any integer , the equation has for small enough a sign-changing solution that develops into a superposition of bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition is not necessary. Indeed, it is known that, if and is a ball , then there is no radial positive solution for small. We complete the picture here by showing that, if , 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 .
"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