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

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 L^p(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 (P_λ) 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.