Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term

Abstract

Motivated by the work [9], in this paper we investigate the infinite boundary value problem associated with the semilinear PDE L⁢u=f⁢(u)+h⁢(x){Lu=f(u)+h(x)} on bounded smooth domains Ω⊆ℝn{\Omega\subseteq\mathbb{R}^{n}}, where L is a non-divergence structure uniformly elliptic operator with singular lower-order terms. In the equation, f is a continuous non-decreasing function that satisfies the Keller–Osserman condition, while h is a continuous function in Ω that may change sign, and which may be unbounded on Ω. Our purpose is two-fold. First we study some sufficient conditions on f and h that would ensure existence of boundary blow-up solutions of the above equation, in which we allow the lower-order coefficients to be singular on the boundary. The second objective is to provide sufficient conditions on f and h for the uniqueness of boundary blow-up solutions. However, to obtain uniqueness, we need the lower-order coefficients of L to be bounded in Ω, but we still allow h to be unbounded on Ω.

Keywords

• large solutions
• existence and uniqueness
• semilinear elliptic equation
• 35j60
• 35j70