Large solutions of elliptic equations with a weakly superlinear nonlinearity |
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Authors: | Florica Corina Cîrstea Yihong Du |
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Institution: | (1) Department of Mathematics, Mathematical Sciences Institute, The Australian National University, Canberra, ACT, 0200, Australia;(2) School of Mathematics, Statistics and Computer Science, University of New England, Armidale, NSW, 2351, Australia;(3) Department of Mathematics, Qufu Normal Univ., P.R. China |
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Abstract: | This paper studies the asymptotic behavior near the boundary for large solutions of the semilinear equation Δu + au = b(x)f(u) in a smooth bounded domain Ω of ℝN with N ≥ 2, where a is a real parameter and b is a nonnegative smooth function on
. We assume that f(u) behaves like u(ln u)α as u → ∞, for some α > 2. It turns out that this case is more difficult to handle than those where f(u) grows like u
p (p > 1) or faster at infinity. Under suitable conditions on the weight function b(x), which may vanish on ∂Ω, we obtain the first order expansion of the large solutions near the boundary. We also obtain some
uniqueness results.
Research of both authors supported by the Australian Research Council. |
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