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October 16, 20250 citationsOpen Access

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

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JMJ. Silverio Martínez-BaenaSVSalvador Villegas

Key Points

  • Semi-stable radial solutions of the Hardy-Henon equation exist in bounded and unbounded forms, offering valuable insights.
  • The established bounds for solutions hold for dimensions between 2 and 10+4α, enhancing understanding of the equation.
  • Sharp pointwise estimates are provided for higher dimensions, improving the theoretical framework for semi-stable solutions.
  • A diverse family of semi-stable unbounded solutions is identified, which could have significant implications in applied contexts.

Abstract

This paper is devoted to the study of semi-stable radial solutions u H¹ (B₁) of -Δu= x^αf (u) in B₁0, where f C¹ (R) is a general nonlinearity, α>-2 and B₁ is the unit ball of RN, N>1. We establish the boudness of such solutions for dimensions 2 N<10+4α and sharp pointwise estimates in the case N10+4α. In addition, we provide, for this range of dimensions, a large family of semi-stable radially decreasing unbounded H¹ (B₁) solutions.

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Cite This Study

Martínez-Baena et al. (2025) studied this question.

synapsesocial.com/papers/68f04acce559138a1a06e99bhttps://doi.org/10.48550/arxiv.2506.21515
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