Elliptic equations with competing rapidly varying nonlinearities and boundary blow-up

We prove that δu + au = b(x)f(u) possesses a unique positive solution such that lim_dist(x,∂ω)→0 u(x) = ∞, where ω is a smooth bounded domain in ℝ^N and a ∈ ℝ. Here b is a smooth function on ω which is positive in ω and may vanish on ∂ω (possibly at a very degenerate rate such as exp(-[dist(x, ∂ω)]^q) with q lt; 0). We assume that f is locally Lipschitz continuous on [0,∞) with f(u)/u increasing for u gt; 0 and f(u) grows at ∞ faster than any power u^p (p gt; 1). As a distinct feature of this study appears the asymptotic behaviour of the boundary blow-up solution, which breaks up depending on how b(x) vanishes on ∂ω and how fast f grows at ∞.