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Abstract. We consider the boundary value problem (P λ ) −∆u = λc(x)u + µ(x)|∇u|where Ω ⊂ R N , N ≥ 3 is a bounded domain with smooth boundary. It is assumed that c 0, c, h belong to L p (Ω) for some p > N . Also µ ∈ L ∞ (Ω) and µ ≥ µ 1 > 0 for some µ 1 ∈ R. It is known that when λ ≤ 0, problem (P λ ) has at most one solution. In this paper we study, under various assumptions, the structure of the set of solutions of (P λ ) assuming that λ > 0. Our study unveils the rich structure of this problem. We show, in particular, that what happen for λ = 0 influences the set of solutions in all the half-space ]0, +∞[×(. Most of our results are valid without assuming that h has a sign. If we require h to have a sign, we observe that the set of solutions differs completely for h 0 and h 0. We also show when h has a sign that solutions not having this sign may exists. Some uniqueness results of signed solutions are also derived. The paper ends with a list of open problems.