Abstract

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as -F(x, u, Du, D(2)u) = lambda c(x)u + (M(x)Du, Du) + h(x) in a bounded domain with a Dirichlet boundary condition; here lambda is an element of R, c, h is an element of L-p(Omega), p > n >= 1, c not greater than or equal to 0 and the matrix M satisfies 0 mu I-1 = M = mu I-2. Recently this problem was studied in the "coercive" case lambda c = 0, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle. We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vazquez type strong maximum principle for our kind of equations. (C) 2018 Elsevier Inc. All rights reserved.