A priori estimates of stable solutions of the general Hardy-Henon equation in the ball

2025/06/26 by J. Silverio Martínez-Baena, Martinez-Baena, J. Silverio, Salvador Villegas +1
Mathematics · #35B35 #35B45 #35J61 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2506.21515

openalex publication_date 2025/06/26 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

This paper is devoted to the study of semi-stable radial solutions u∈ H¹(B₁) of -Δu=\vert x\vert^αf(u) in B₁∖\lbrace0\rbrace, where f∈ C¹(ℝ) is a general nonlinearity, α>-2 and B₁ is the unit ball of ℝ^N, N>1. We establish the boudness of such solutions for dimensions 2≤ N<10+4α and sharp pointwise estimates in the case N≥10+4α. In addition, we provide, for this range of dimensions, a large family of semi-stable radially decreasing unbounded H¹(B₁) solutions.

Citations

• Partial regularity of stable solutions to the fractional Gel'fand-Liouville equation
• On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand-Liouville equation
• On the fractional Lane-Emden equation
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

Related

• Sharp estimates for semi-stable radial solutions of semilinear elliptic equations
• Dichotomy of stable radial solutions of -Δu=f(u) outside a ball
• Non-energy semi-stable radial solutions
• Singularity and blow-up estimates via Liouville-type theorems for Hardy-Hénon parabolic equations
• Asymptotic profile and Morse index of the radial solutions of the Hénon equation