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

Colette De Coster (1) , Louis Jeanjean (2)

1 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

We consider the boundary value problem (Pλ) −∆u = λc(x)u + µ(x)|∇u| 2 + h(x), u ∈ H1 0 (Ω) ∩ L ∞(Ω), 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 ≥ 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, +∞[×(H1 0 (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

Domaines: Equations aux dérivées partielles [math.AP]

Colette De Coster, Louis Jeanjean. Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient. Journal of Differential Equations, 2017, 262 (10), pp.5231 - 5270. ⟨10.1016/j.jde.2017.01.022⟩. ⟨hal-01755985⟩