Quasilinear problems involving a perturbation with quadratic growth in the gradient and a noncoercive zeroth order term

Abstract

In this paper we consider the problem u in H^1_0 (Omega), - div (A(x) Du) = H(x, u, Du) + f(x) + a_0 (x) u in D'(Omega), where Omega is an open bounded set of R^N, N \geq 3, A(x) is a coercive matrix with coefficients in L^\infty(Omega), H(x, s, xi) is a Carath\'eodory function which satisfies for some gamma > 0 -c_0 A(x) xi xi \leq H(x, s, xi) sign (s) \leq gamma A(x) xi xi a.e. x in Omega, forall s in R, forall xi in R^N, f belongs to L^{N/2} (Omega), and a_0 \geq 0 to L^q (Omega ), q > N/2. For f and a_0 sufficiently small, we prove the existence of at least one solution u of this problem which is moreover such that e^{delta_0 |u|} - 1 belongs to H^1_0 (Omega) for some delta_0 \geq gamma, and which satisfies an a priori estimate.