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Article Dans Une Revue Journal of Functional Analysis Année : 2015

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

Résumé

We consider the boundary value problem(Pλ)u∈H01(Ω)∩L∞(Ω):-δu=λc(x)u+μ(x)|∇u|2+h(x), where Ω⊂RN, N≥3 is a bounded domain with smooth boundary. It is assumed that c{greater-than but not equal to}0, c, h belong to Lp(Ω) for some p>N/2 and that μ∈L∞(Ω). We explicitly describe a condition which guarantees the existence of a unique solution of (Pλ) when λ<0 and we show that these solutions belong to a continuum. The behaviour of the continuum depends in an essential way on the existence of a solution of (P0). It crosses the axis λ=0 if (P0) has a solution, otherwise it bifurcates from infinity at the left of the axis λ=0. Assuming that (P0) has a solution and strengthening our assumptions to μ(x)≥μ1>0 and h{greater-than but not equal to}0, we show that the continuum bifurcates from infinity on the right of the axis λ=0 and this implies, in particular, the existence of two solutions for any λ>0 sufficiently small.

Dates et versions

hal-01755994 , version 1 (31-03-2018)

Identifiants

Citer

David Arcoya, Colette De Coster, Louis Jeanjean, Kazunaga Tanaka. Continuum of solutions for an elliptic problem with critical growth in the gradient. Journal of Functional Analysis, 2015, 268 (8), pp.2298 - 2335. ⟨10.1016/j.jfa.2015.01.014⟩. ⟨hal-01755994⟩
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