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

We consider the boundary value problem -Delta u = lambda c(x)u + mu(x)vertical bar del u vertical bar(2) + h(x), u epsilon H-0(1) (Omega) boolean AND L-infinity (Omega), (P-lambda) where Omega subset of R-N, N >= 3 is a bounded domain with smooth boundary. It is assumed that c, h belong to L-P (Omega) for some p > N with c(() (>)(sic)) 0 as well as mu epsilon L-infinity (Omega) and mu >= mu(1) > 0 for some mu(1) epsilon R. It is known that when lambda <= 0, problem (P-lambda) has at most one solution. In this paper we study, under various assumptions, the structure of the set of solutions of (P-lambda) assuming that lambda > 0. Our study unveils the rich structure of this problem. We show, in particular, that what happen for lambda = 0 influences the set of solutions in all the half-space]0, +infinity[x (H-0(1) (Omega) boolean AND L-infinity (Omega)). 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 (( sic)) 0 and h (( sic)) 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. (C) 2017 Elsevier Inc. All rights reserved.