Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 130, pp. 1–9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NON-EXISTENCE OF POSITIVE RADIAL SOLUTION FOR SEMIPOSITONE WEIGHTED P-LAPLACIAN PROBLEMS SIGIFREDO HERR ´ ON, EMER LOPERA Abstract. We prove the non-existence of positive radial solution to a semi- positone weighted p-Laplacian problem whenever the weight is sufficiently large. Our main tools are a Pohozaev type identity and a comparison principle. 1. Introduction We consider the non-existence of positive radial solutions to the problem −Δ p u = W (‖x‖)f (u) in B 1 (0), u =0 on ∂B 1 (0), (1.1) where Δ p u = ∇· (|∇u| p−2 ∇u) is the p-Laplace operator, B 1 (0) is the unit ball in R N and 2 <p<N . Note that solving this problem is equivalent to solving the problem [r n ϕ p (u ′ )] ′ = −r n W (r)f (u), 0 <r< 1, u ′ (0) = 0, u(1) = 0, (1.2) where r = ‖x‖, n := N − 1 and ′ = d dr . The differential equation in the last problem is equivalent to (p − 1)|u ′ | p−2 u ′′ + n r |u ′ | p−2 u ′ + W (r)f (u)=0, 0 <r< 1. (1.3) We assume that the nonlinearity satisfies the following hypotheses: (F1) f : [0, ∞) → R is a continuous function with exactly three zeros 0 <β 1 < β 2 <β 3 . (F2) f (0) < 0 and f is increasing from β 3 on. (F3) Set F (t) :=  t 0 f (s)ds. Then, β 3 <θ 1 where θ 1 is the unique positive zero of F . Let us fix β 3 <γ<θ 1 . We will say that a function W is an admissible weight if it satisfies the following conditions: (W1) W : [0, 1] → (0, ∞) is continuous and differentiable in (0, 1). (W2)  W (r) := N + r W ′ (r) W (r) is defined a.e. in [0, 1]. 2010 Mathematics Subject Classification. 35J92, 35J60, 35J62. Key words and phrases. Semipositone; quasilinear weighted elliptic equation; positive radial solution; non-existence result; Pohozaev identity. c 2015 Texas State University - San Marcos. Submitted January 12, 2015. Published May 7, 2015. 1