On the \(Q\)-curvature problem on \(\mathbb S^3\) (Q2832813)
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scientific article; zbMATH DE number 6652849
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the \(Q\)-curvature problem on \(\mathbb S^3\) |
scientific article; zbMATH DE number 6652849 |
Statements
14 November 2016
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fourth order elliptic equations on manifold
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negative exponent
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existence
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degree theory
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positive solutions
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On the \(Q\)-curvature problem on \(\mathbb S^3\) (English)
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Let \((\mathbb{S}^3,g_0)\) be the standard three sphere and \(\Delta_0\) the Laplace-Beltrami operator on it. The authors prove existence of positive solutions \(u\) to the equation NEWLINE\[NEWLINE \Delta^2_0u+\dfrac{1}{2}\Delta_0u-\dfrac{15}{16}u+\dfrac{1}{2}Qu^{-7}=0\quad \text{on}\;\mathbb{S}^3, NEWLINE\]NEWLINE where \(Q\) is a smooth and positive function, satisfying the non-degeneracy condition NEWLINE\[NEWLINE (\Delta_0Q(x))^2+|\nabla Q(x)|^2\neq 0\quad \forall x\in \mathbb{S}^3. NEWLINE\]NEWLINE The proof of the existence result relies on variational arguments and the Leray-Schauder degree theory.
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