The existence and boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms

Abstract

In this paper, for more general f, g and a, b, we obtain conditions about the existence and boundary behavior of solutions to boundary blow-up elliptic problems ▵u=a(x)g(u)+b(x)f(u)|∇u|q,x∈Ω,u|∂Ω=+∞$ \triangle u=a(x)g(u)+ b(x) f(u)|\nabla u|^q,\quad x\in \Omega ,\quad u|_{\partial \Omega }=+\infty $ and improve and generalize most of the previously available results in the literature, where Ω is a bounded domain with smooth boundary in ℝN, q∈(0,2]${q\in (0, 2]}$, a,b∈Cν(Ω¯)${a, b \in C^{\nu }(\overline{\Omega })}$ which are positive in Ω, may be vanishing on the boundary, and f,g∈C[0,∞)∩C1(0,∞)${f, g\in C[0, \infty ) \cap C^1(0, \infty ) }$ or f,g∈C1(ℝ)${f, g\in C^1(\mathbb {R}) }$, which are increasing.

Keywords

• semilinear elliptic equations
• nonlinear gradient terms
• boundary blow-up
• 35j15
• 35j25
• 35j61