Gradient estimate for solutions of a class of nonlinear elliptic equations below the duality exponent

We study the regularity of a solution of the Dirichlet problem associated to the singular equation | Du|2 −div(a(x)Du)+M uθ = f(x) inΩ (1) where Ω is an open bounded subset of RN (N ≥ 3) with smooth boundary, a(x) is a L∞–matrix satisfying the standard ellipticity condition, θ ∈]0, 1[, M is a positive constant and f is sufficiently regular i.e. it belongs to a suitable Morrey space Lq,λ (Ω), with q ≥ 1, to be specified later on. We will be concerned with the regularity of the gradient of a solution in Morrey spaces in correspondence with the regularity properties of the right–hand side of the equation (1). There is a huge literature about the problems with quadratic term in the gradient also for high-order equations whose coefficients satisfy a strengthened ellipticity condition (see [5]). The problem (1) has been studied in the paper [2] by D. Arcoya, J. Carmona, T. Leonori, P. J. Mart ́ınez-Aparicio, L. Orsina, F. Petitta and in the paper [3] by L. Boccardo where the source term f belonged to Lq(Ω) with q ≥ 1. In [4] we have extended to the gradient of a solution the Morrey property of the right–hand side f and, in some cases, we have improved some results contained in [2, 3] without increasing the summability of f . Here we will be concerned with an intermediate case, in the sense that the right hand side is not merely a measure and it doesn’t belong to the right dual space (see [10, 9]). In obtaining the necessary local estimates we faced the problem of performing the Campanato’s decomposition of the solution due to the presence of the degenerate lower order term. We had thus to adopt an alternative method already used in [4]. This is part of a new set of estimates, including potential estimates, as for instance in [6, 7, 8], that would be very interesting to extend in the present setting too.