Multiple positive solutions for a singular biharmonic equation in ℝ⁴

Abstract

Let B ₁(0) ⊂ ℝ⁴ be the unit ball. In this article, we study the existence of multiple positive solutions of the following biharmonic problem:

where 0 < q < 1 < α ≤ 2, λ ∈ ℝ, 0 < β < 4 and h satisfies certain growth conditions mentioned in Section 1. We show that there exists 0 < Λ < ∞ such that the above problem admits at least two solutions in if λ ∈ (0, Λ), no solution if λ > Λ and at least one solution when λ = Λ.

Keywords:

• biharmonic equation
• multiple solutions

AMS Subject Classifications:

• 35J35
• 35J40
• 35J70

Acknowledgements

Part of this work was done when the first author was visiting the Université de Pau et des Pays de l’Adour. She would like to thank Professor J. Giacomoni for encouraging discussions on this topic.