We consider the following perturbed critical Dirichlet problem involving the Hardy–Schrödinger operator: (Formula presented) when ε > 0 is small, (Formula presented), and where (Formula presented), N ≤ 3, is a smooth bounded domain with 0 ∈ Ω. We show that there exists a sequence (Formula presented) with (Formula presented) such that, if (Formula presented) for any j and (Formula presented), then the above equation has for ε small, a positive — in general nonminimizing — solution that develops a bubble at the origin. If moreover (Formula presented), then for any integer k ≤ 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 (Formula presented) is not necessary. Indeed, it is known that, if (Formula presented) 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 (Formula presented), 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 γ = 0.

Sign-changing solutions for critical equations with hardy potential

Esposito P.;Vaira G.
2021-01-01

Abstract

We consider the following perturbed critical Dirichlet problem involving the Hardy–Schrödinger operator: (Formula presented) when ε > 0 is small, (Formula presented), and where (Formula presented), N ≤ 3, is a smooth bounded domain with 0 ∈ Ω. We show that there exists a sequence (Formula presented) with (Formula presented) such that, if (Formula presented) for any j and (Formula presented), then the above equation has for ε small, a positive — in general nonminimizing — solution that develops a bubble at the origin. If moreover (Formula presented), then for any integer k ≤ 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 (Formula presented) is not necessary. Indeed, it is known that, if (Formula presented) 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 (Formula presented), 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 γ = 0.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11586/379881
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