Boundary blow-up solutions to singular k-Hessian equations with gradient terms: Sufficient and necessary conditions and asymptotic behavior

Abstract

Let Ω be a smooth, bounded and strictly (k−1)-convex domain in RN(N≥2). Assume that b and f are smooth positive functions and b may be singular near ∂Ω. We provide sufficient and necessary conditions on b, f and q for the existence of k-convex solutions to the boundary blow-up k-Hessian problemSk(D2u)=b(x)f(u)|∇u|q for x∈Ω,u(x)→+∞ as dist(x,∂Ω)→0. The asymptotic behavior of such solutions is also investigated. Moreover, when b has strong singularity near ∂Ω, we study the existence and nonexistence of boundary blow-up solutions.