The continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains

Abstract

The goal of this paper is to discuss the continuous dependence of solutions on functional parameters for the following semilinear elliptic partial differential equation: Δ u ( x ) + f ¯ ( x , u ( x ) , v ( ‖ x ‖ ) ) + g ( ‖ x ‖ ) x ⋅ ∇ u ( x ) = 0 , for x ∈ Ω r 0 ≔ { x ∈ R n , n ≥ 3 , ‖ x ‖ > r 0 } and v ∈ V , where V stands for some functional space. Our approach covers the case when f may change sign and admits general growth. As an additional result, the characterization of the radius r 0 for which our problem possesses at least one positive evanescent solution in the exterior domain Ω r 0 is described and numerically illustrated. Our approach relies on the subsolution and supersolution method and on a lemma due to Noussair and Swanson.