Continuum of solutions for an elliptic problem with critical growth in the gradient

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

“…When λ = 0 this was already observed in [12]. By [3], the uniqueness itself holds as soon as λc(x) ≤ 0 a.e. in Ω.…”

“…In addition this negative solution is unique [2,Theorem 3.12]. Considered together, the results of [2,3] show that the sign of h has definitely an influence on the set of solutions of (P λ ) when λ > 0.…”

“…There had been a lot of contributions [1,12,17,19] when λ = 0 (or equivalently when c ≡ 0) but the general case where λc ≤ 0 may vanish only on some parts of Ω was left open until the paper [3]. It appears in [3] that under assumption (A) the existence of solutions is not guaranteed, additional conditions are necessary. When λ = 0 this was already observed in [12].…”

“…This result have been complemented in [15] where two solutions are obtained, allowing the function c to change sign but assuming that h ≥ 0 and that max{0, λc} 0. The restriction that µ is a constant was subsequently removed in [3] under the price of the assumption h ≥ 0.…”

“…In [3,Lemma 6.1], letting γ 1 > 0 be the first eigenvalue of (1.1) − ∆ϕ 1 = γc(x)ϕ 1 , ϕ 1 ∈ H 1 0 (Ω), it is proved when h ≥ 0 that problem (P λ ) has no solution when λ = γ 1 and no non-negative solutions when λ > γ 1 . This contrasts to what was observed in [2,Theorem 3.3], namely that if µ > 0 is a constant and h 0, then there exists a negative solution of (P λ ) as soon as λ > 0.…”