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Exact boundary behavior of large solutions to semilinear elliptic equations with a nonlinear gradient term

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Abstract

This paper is concerned with exact boundary behavior of large solutions to semilinear elliptic equations Δu = b(x)f(u)(C0 + |∇u|q), x ∈ Ω, where Ω is a bounded domain with a smooth boundary in ℝN, C0 ⩾ 0, q ∈ [0, 2), \(b \in C_{\rm{loc}}^\alpha(\Omega)\) is positive in Ω, but may be vanishing or appropriately singular on the boundary, f ∈ C[0,∞), f(0) = 0, and f is strictly increasing on [0, ∞) (or f ∈ C(ℝ), f(s) > ∀ s ∈ ℝ, f is strictly increasing on ℝ). We show unified boundary behavior of such solutions to the problem under a new structure condition on f.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11571295). The author is greatly indebted to the anonymous referees for the very helpful suggestions and com- ments which improved the quality of the presentation.

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  1. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, China

    Zhijun Zhang

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  1. Zhijun Zhang
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Zhang, Z. Exact boundary behavior of large solutions to semilinear elliptic equations with a nonlinear gradient term. Sci. China Math. 63, 559–574 (2020). https://doi.org/10.1007/s11425-017-9275-y

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