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

Abstract 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 problem S k ( D 2 u ( 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 } , S k ( D 2 u ) is the k-Hessian operator, β > 0 and Ω is a smooth, bounded, strictly convex domain in R N ( N ≥ 2 ) . We suppose that the nonlinearity behaves like u k ln β ⁡ u as u → ∞ , which is more complex and difficult to deal with than the nonlinearity grows like u p 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.