Boundary blow-up solutions to the k-Hessian equation with a weakly superlinear nonlinearity

By constructing new sub- and super-solutions, we are concerned with determining values of β, for which there exist k-convex solutions to the boundary blow-up k-Hessian problemSk(D2u(x))=H(x)[u(x)]k[ln⁡u(x)]β>0 for x∈Ω,u(x)→+∞ as dist(x,∂Ω)→0. Here k∈{1,2,⋯,N}, Sk(D2u) is the k-Hessian operator, β>0 and Ω is a smooth, bounded, strictly convex domain in RN(N≥2). We suppose that the nonlinearity behaves like uklnβ⁡u as u→∞, which is more complex and difficult to deal with than the nonlinearity grows like up with p>k or faster at infinity. Further, several new results of nonexistence, global estimates and estimates near the boundary for the solutions are also given.