On a k-Hessian equation with a weakly superlinear nonlinearity and singular weights. (January 2020) - Feng, Meiqiang; Zhang, Xuemei
Abstract: Consider the existence, nonexistence and global estimates of 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, Ω is a smooth, bounded, strictly convex domain in R N ( N ≥ 2 ), and H ( x ) is a positive weight function which is singular near the boundary ∂ Ω . We first give the existence and nonexistence results of k -convex solution to the above boundary blow-up problem on a larger range of H and β . Then we show that there is a k -convex solution provided that H ( x ) grows fast near ∂ Ω and u k [ ln u ] β grows slow at ∞ . It turns out that this case is more difficult to handle than the case in which H ( x ) grows slow near ∂ Ω and u k [ ln u ] β grows fast at ∞ . This needs some new ingredients in the arguments.