Existence and regularity results for fully nonlinear equations with singularities

Abstract

In this paper we consider the Dirichlet boundary value problem for a singular elliptic PDE lke F [u] = p(x)u −µ , where p, µ ≥ 0, in a bounded domain in R n . The nondivergence form operator F is assumed to be of Hamilton-Jacobi-Bellman or Isaacs type. We establish existence and regularity results for such equations.

Positive solution to extremal Pucci's equations with singular and gradient nonlinearity

“…This works extend and complement the earlier research works on singular elliptic equations due to the following: (a) In case q = 1, where q is given next in (H2), and f (t) ≡ 1, the existence and regularity properties of positive solution to even more general problem than (1) has been studied in [23]. Thus, our results extend the work of [23] because here, we deal with the problem which has superlinear growth in the gradient as well as we allow an additional nonlinear term f. (b) In case α = 0 and either q = 1 or H ≡ 0, our operator reduces to proper operator. The existence of solutions to these type of problems been studied in [18,17].…”

“…Thus g τ → g in L p (Ω) for all p ∈ [1, ∞). Since v τ satisfies (73) so by an application of the stability result (Theorem 5 [23]) for viscosity solution, we find that v satisfies the following equation…”

“…For the complete list of references in this direction, we refer to survey [31]. In the context of fully nonlinear elliptic equations such problems have been considered in [23], where authors studied the existence and regularity properties of the solutions. Recently, the existence of positive solution to Pucci's extremal equation involving singular and sublinear nonlinearity has been established in [46].…”

The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity

“…On the other hand less is known in the fully nonlinear setting. In [8] the authors extend the existence and regularity results of solutions as in the semilinear case to Hamilton-Jacobi-Bellman and Isaacs uniformly elliptic operators. As far as we know there are no works dealing with pure degenerate elliptic equations of fully nonlinear type.…”

Geometric Approach to Nonvariational Singular Elliptic Equations

“…Notice that such a problem is quite different from the one treated in the classical paper [11]. The latter has been recently studied in the fully nonlinear setting in [13].…”