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Uniqueness of large positive solutions

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Abstract

We establish the uniqueness of the positive solution of the singular problem (1.1) through some standard comparison techniques involving the maximum principle. Our proofs do not invoke to the blow-up rates of the solutions, as in most of the specialized literature. We give two different types of results according to the geometrical properties of \(\Omega \) and the regularity of \(\partial \Omega \). Even in the autonomous case, our theorems are extremely sharp extensions of all existing results. Precisely, when \(\mathfrak {a}(x)\equiv 1\), it is shown that the monotonicity and superadditivity of f(u) with constant \(C\ge 0\) entail the uniqueness; f is said to be superadditive with constant \(C\ge 0\) if

$$\begin{aligned} f(a+b) \ge f(a) + f(b) - C \quad \hbox {for all}\;\; a, b \ge 0. \end{aligned}$$

This condition, introduced by Marcus and Véron (J Evol Equ 3:637–652, 2004), weakens all previous sufficient conditions for uniqueness, as it will become apparent in this paper.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, 28040, Madrid, Spain

    Julián López-Gómez

  2. Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040, Madrid, Spain

    Luis Maire

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  1. Julián López-Gómez
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  2. Luis Maire
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Correspondence to Luis Maire.

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Supported by the Ministry of Economy and Competitiveness under Research Grant MTM2015-65899-P.

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López-Gómez, J., Maire, L. Uniqueness of large positive solutions. Z. Angew. Math. Phys. 68, 86 (2017). https://doi.org/10.1007/s00033-017-0829-1

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  • DOI: https://doi.org/10.1007/s00033-017-0829-1

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