Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient

We consider the boundary value problem(Pλ)−Δu=λc(x)u+μ(x)|∇u|2+h(x),u∈H01(Ω)∩L∞(Ω), where Ω⊂RN,N≥3 is a bounded domain with smooth boundary. It is assumed that c,h belong to Lp(Ω) for some p>N with c≩0 as well as μ∈L∞(Ω) and μ≥μ1>0 for some μ1∈R. It is known that when λ≤0, problem (Pλ) has at most one solution. In this paper we study, under various assumptions, the structure of the set of solutions of (Pλ) 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,+∞[×(H01(Ω)∩L∞(Ω)). 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.