Classes of reaction diffusion equations where a parameter influences the equation as well as the boundary condition

Abstract We study positive solutions to steady state reaction diffusion equations of the form: { − Δ u = λ f ( u ) ; Ω , ∂ u ∂ η + μ ( λ ) u = 0 ; ∂ Ω , where λ > 0 , Ω is a bounded domain in R N ; N ≥ 1 with smooth boundary ∂Ω, ∂ u ∂ η is the outward normal derivative of u , μ ∈ C ( [ 0 , ∞ ) ) is strictly increasing such that μ ( 0 ) ≥ 0 and f ∈ C 2 ( [ 0 , r 0 ) ) with 0 r 0 ≤ ∞ . If r 0 ∞ we assume f ∈ C 2 ( [ 0 , r 0 ] ) with f ( r 0 ) = 0 and f ( s ) ≤ 0 for s ∈ ( r 0 , ∞ ) , and if r 0 = ∞ we assume lim s → ∞ ⁡ f ( s ) > 0 and lim s → ∞ ⁡ f ( s ) s = 0 (sublinear at ∞). Note here that the parameter λ influences both the equation and the boundary condition. We discuss existence, nonexistence, multiplicity and uniqueness results for the cases when (A) f ( 0 ) = 0 , (B) f ( 0 ) 0 , and (C) f ( 0 ) > 0 . We obtain existence and multiplicity results by the method of sub-super solutions and uniqueness results by comparison principles and a priori estimates.