We consider a quasilinear elliptic problem of the form
where λ > 0 is a parameter, 1 < p < 2 and Ω is a strictly convex bounded domain in ℝ
, N > p, with C
2 boundary ∂Ω. The nonlinearity f : [0, ∞) → ℝ is a continuous function that is semipositone (f(0) < 0) and p-superlinear at infinity. Using degree theory, combined with a rescaling argument and uniform L
a priori bound, we establish the existence of a positive solution for λ small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at λ = 0. We also extend our study to systems.