MULTIPLICITY RESULTS IN THE NON-COERCIVE CASE FOR AN ELLIPTIC PROBLEM WITH CRITICAL GROWTH IN THE GRADIENT

Abstract

Abstract. We consider the boundary value problem (Pλ) −∆u = λc(x)u+ µ(x)|∇u|2 + h(x), u ∈ H10 (Ω) ∩ L∞(Ω), where Ω ⊂ RN, N ≥ 3 is a bounded domain with smooth boundary. It is assumed that c \t 0, c, h belong to Lp(Ω) for some p> N. Also 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,+∞[×(H10 (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 \t 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. 1.