Abstract
We prove that δu + au = b(x)f(u) possesses a unique positive solution such that limdist(x,∂ω)→0 u(x) = ∞, where ω is a smooth bounded domain in ℝN 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 up (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 ∞.
Original language | English |
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Pages (from-to) | 995-1030 |
Number of pages | 36 |
Journal | Advances in Differential Equations |
Volume | 12 |
Issue number | 9 |
Publication status | Published - 2007 |