A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient

We study nonnegative solutions of the boundary value problem equation(PλPλ)−Δu=λc(x)u+μ(x)∣∇u∣2+h(x),u∈H01(Ω)∩L∞(Ω), where ΩΩ is a smooth bounded domain of RnRn, μ,c∈L∞(Ω)μ,c∈L∞(Ω), h∈Lr(Ω)h∈Lr(Ω) for some r>n/2r>n/2 and μ,c,h≩0μ,c,h≩0. Our main motivation is to study the “semidefinite” case. Namely, unlike in previous work on the subject, we do not assume μμ to be uniformly positive in ΩΩ, nor even positive everywhere.In space dimensions up to n=5n=5, we establish uniform a priori estimates for weak solutions of (PλPλ) when λ>0λ>0 is bounded away from 00. This is proved under the assumption that the supports of μμ and cc intersect, a condition that we show to be actually necessary, and in some cases we further assume that μμ is uniformly positive on the support of cc and/or some other conditions.As a consequence of our a priori estimates, assuming that (P0P0) has a solution, we deduce the existence of a continuum CC of solutions, such that the projection of CC onto the λλ-axis is an interval of the form [0,a][0,a] for some a>0a>0 and that the continuum CC bifurcates from infinity to the right of the axis λ=0λ=0. In particular, for each λ>0λ>0 small enough, problem (Pλ)(Pλ) has at least two distinct solutions.