A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient

Let Omega subset of R-N, N >= 2, be a smooth bounded domain. We consider a boundary value problem of the form -Delta u = c(lambda)(x)u + mu(x)vertical bar del u vertical bar(2) + h(x), u is an element of H-0(1)(Omega) boolean AND L-infinity(Omega) where c(lambda) depends on a parameter lambda is an element of R, the coefficients c(lambda) and h belong to L-q (Omega) with q > N/2 and mu is an element of L-infinity(Omega). Under suitable assumptions, but without imposing a sign condition on any of these coefficients, we obtain an a priori upper bound on the solutions. Our proof relies on a new boundary weak Harnack inequality. This inequality, which is of independent interest, is established in the general framework of the p-Laplacian. With this a priori bound at hand, we show the existence and multiplicity of solutions. (C) 2019 Elsevier Masson SAS. All rights reserved.