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

`We consider the boundary value problem u is an element of H-0(1)(Omega) boolean AND L-infinity(Omega) : -Delta u = lambda c(x)u + mu(x)vertical bar Delta u vertical bar(2) + h(x) where Omega subset of R-N, N >= 3 is a bounded domain with smooth boundary. It is assumed that c not greater than or equal to 0, c, h belong to L-P (Omega) for some p > N/2 and that it mu is an element of L-infinity (Omega). We explicitly describe a condition which guarantees the existence of a unique solution of (P-lambda) when lambda < 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 (Po).It crosses the axis lambda = 0 if (P-0) has a solution, otherwise it bifurcates from infinity at the left Of the axis lambda = 0. Assuming that (P-0) has a solution and strengthening our assumptions to u(x) >= mu 11 > 0 and h not greater than or equal to 0, we show that the continuum bifurcates from infinity on the right of the axis A = 0 and this implies, in particular, the existence of two solutions for any lambda > 0 sufficiently small. (C) 2015 Elsevier Inc. All rights reserved.