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

Abstract

In this paper, we establish the existence of a positive solution to        − M + λ,Λ (D 2 u) + H(x, Du) = k(x)f (u) u α in Ω, u > 0 in Ω, u = 0 on ∂Ω, under certain conditions on k, f and H, using viscosity sub-and supersolution method. The main feature of this problem is that it has singularity as well as a superlinear growth in the gradient term. We use Hopf-Cole transformation to handle the superlinear gradient term and an approximation method combined with suitable stability result for viscosity solution to outfit the singular nonlinearity. This work extends and complements the recent works on elliptic equations involving singular as well as superlinear gradient nonlinearities.

Fully nonlinear degenerate equations with applications to Grad equations

“…We refer to [10,11,18,20,21,23,24,25] for the existence and qualitative questions pertaining to extremal Pucci's equations. On the other hand, equations concerning gradient degenerate fully nonlinear elliptic operators have been investigated widely in the last decade.…”

Lyapunov-Type Inequality for Extremal Pucci’s Equations

“…For the Lyapunov-type inequality for the p-Laplace equation with the Dirichlet boundary condition, we refer to [12]. Motivated by the recent works on the existence and qualitative questions for fully nonlinear elliptic equations and, in particular, extremal Pucci's equations, see for instance [9,30,[32][33][34], the following is a natural question to ask. [4] Question.…”

Regional Dynamics of Rural Credit and Growth in India: Exploring Nonlinearity and Convergence in Growth Patterns

“…The existence of solution to (1.1) with sublinear nonlinear term as well as associated semipositone problems have been established in [27]. Although the existence of solution to a class of equations having superlinear growth in gradient and singular non linear term is proved in [28] but the regularity of the solutions to such equations has not been proved yet. The equations having superlinear growth in gradient have been attaracting continuous attention of the researchers.…”