Global structure of positive solutions for a singular quasilinear elliptic problem

Abstract

We are concerned with the existence of positive solutions for the quasilinear problem (P) { − Δ p u = λK ( | x | ) f ( u ) , x ∈ R N ∖ B ( r 0 ) , g ~ 1 ( u ) u − a 1 ∇u ⋅ x | x | → 0 , | x | → ∞ , g ~ 2 ( u ) u + a 2 ∂ u ∂ n = 0 , x ∈ ∂B ( r 0 ) , where Δ p s = div ( | ∇s | p − 2 ⋅ ∇s ) , 1<p<N, 0 $ ]]> λ > 0 is a parameter, 0 $ ]]> a i , r 0 > 0 are constants for i = 1, 2, B ( r 0 ) := { x ∈ R N : | x | < r 0 } , ∂ u ∂ n is the outward normal derivative of u on ∂B ( r 0 ) , K : ( r 0 , ∞ ) → ( 0 , ∞ ) is a continuous function, f : ( 0 , ∞ ) → R is a continuous function which satisfies lim s → ∞ f ( s ) / φ p ( s ) = ∞ , g ~ i : [ 0 , ∞ ) → ( 0 , ∞ ) are continuous functions. We investigate the global structure of positive solution for (P). The proof of main result is based upon bifurcation theory.