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Large solutions of elliptic equations with a weakly superlinear nonlinearity

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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 \(\overline \Omega \). 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.

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Authors and Affiliations

  1. Department of Mathematics, Mathematical Sciences Institute, The Australian National University, Canberra, ACT, 0200, Australia

    Florica Corina Cîrstea

  2. School of Mathematics, Statistics and Computer Science, University of New England, Armidale, NSW, 2351, Australia

    Yihong Du

  3. Department of Mathematics, Qufu Normal Univ., P.R. China

    Yihong Du

Authors
  1. Florica Corina Cîrstea
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  2. Yihong Du
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Correspondence to Florica Corina Cîrstea.

Additional information

Research of both authors supported by the Australian Research Council.

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Cîrstea, F.C., Du, Y. Large solutions of elliptic equations with a weakly superlinear nonlinearity. J Anal Math 103, 261–277 (2007). https://doi.org/10.1007/s11854-008-0008-6

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  • DOI: https://doi.org/10.1007/s11854-008-0008-6

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