Positive radial solutions to singular nonlinear elliptic problems involving nonhomogeneous operators. (March 2022) - Sim, Inbo; Son, Byungjae
Abstract: We consider singular nonlinear elliptic problems involving nonhomogeneous operators on annular domains − div ( A ( | x | ) B ( | ∇ u | ) ∇ u ) = λ K ( | x | ) f ( u ), x ∈ Ω, u = 0, | x | = r 1, a ∂ u ∂ n + c ( λ, u ) = 0, | x | = r 2, where λ > 0, a ≥ 0, N > 1, Ω ≔ { x ∈ R N ∣ 0 < r 1 < | x | < r 2 < ∞ } and ∂ u ∂ n is the outward normal derivative of u on ∂ B r 2 . Here A ∈ C ( [ r 1, r 2 ], ( 0, ∞ ) ), K ∈ C ( ( r 1, r 2 ), ( 0, ∞ ) ), c ∈ C ( ( 0, ∞ ) × R, R ), B ∈ C ( [ 0, ∞ ), [ 0, ∞ ) ) is such that B ( s ) s is a homeomorphism from [ 0, ∞ ) onto [ 0, ∞ ), and f ∈ C ( ( 0, ∞ ), ( 0, ∞ ) ) has a singularity at 0. The aim of this paper is to analyze the existence and multiplicity of positive radial solutions according to the behavior of f near ∞ . In particular, we discuss sufficient conditions for at least three positive radial solutions to exist. The results are obtained via a Krasnoselskii type fixed point theorem. Finally, we provide examples including Gelfand-type problems to illustrate each result.