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

Advances in Nonlinear Analysis

article

Ahmed Mohammed¹  Giovanni Porru² 

[1] Department of Mathematical Sciences, Ball State University;Dipartimento di Matematica e Informatica, University of Cagliari

关键词: Large solutions;    existence and uniqueness;    semilinear elliptic equation;   

DOI  :  10.1515/anona-2017-0065

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

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 ⊆ ℝ n {\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.

【 授权许可】

CC BY