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Clustered solutions to low-order perturbations of fractional Yamabe equations

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Abstract

Let \((X, g^+)\) be an asymptotically hyperbolic manifold and \((M, [\hat{h}])\) be its conformal infinity. We construct positive clustered solutions to low-order perturbations of \(\gamma \)-Yamabe equations (\(0< \gamma < 1\)) on \((M, \hat{h})\), which are slightly supercritical, under certain geometric and dimensional assumptions. These solutions certainly exhibit non-isolated blow-up.

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Notes

  1. The existence and uniqueness of the solution \(U \in H^{1,2}(X; \rho ^{1-2\gamma })\) for any given function \(g \in L^{p+1}(M)\) is guaranteed by the definition of the *-norm in (2.10) and the Riesz representation theorem. Also, [10, Lemma 3.3] implies that if \(g \in L^{q_{\epsilon }/(p+\epsilon )}(M)\), then \(u = U|_M \in L^{q_{\epsilon }}(M)\).

  2. By (2.8), it holds that \(\delta ^{2\gamma } \sim \epsilon \), \(\delta ^{(n-2\gamma ) (\alpha -\beta )} \sim \epsilon ^{1 + {n-4\gamma \over \gamma (n-2\gamma +2)}}\) and \(\delta ^2 \sim \epsilon ^{1 \over \gamma }\).

  3. The arguments in [25, Section 2] involving the Moser iteration technique also work in our setting, see [18, Section 3]. Also, in order to take advantage of the regularity theory, we need that the value of \(\sup _{\epsilon > 0} \Vert \Phi _{\ell \epsilon }^{p-1+\epsilon }\Vert _{L^{n/(2\gamma )}(M \setminus B_{\hat{h}}(\xi _0, r))}\) is sufficiently small for a fixed radius \(r > 0\), which is true owing to our choice of \(q_{\epsilon }\) in (2.11).

  4. Once it is proved, even though the maximization argument does not work, we may apply the general critical point result of Thizy and Vétois [46, Lemma A.1].

  5. Precisely, \(\mathcal {M}_0 > 0\) if \(n \ge 4\) for \(\gamma \in (\sqrt{5/11}, 1)\), \(n \ge 5\) for \(\gamma \in (1/2, 1)\), \(n \ge 6\) for \(\gamma \in (\sqrt{1/19}, 1)\) and \(n \ge 7\) for any \(\gamma \in (0,1)\). Observe that \(\mathcal {M}_0\) is the quantity that already appeared in [29, Corollary 2.7].

  6. Here \(\zeta > 0\) is an arbitrary number in the nonempty interval \(((2\gamma n-4\gamma ^2-4)/(n-2\gamma +2), 2\gamma ]\) such that \(n > 2+2\gamma +\zeta \) (cf. Proof of Corollary 3.7 (1)).

  7. The second integral in the parenthesis is \(O(\epsilon ^{1/\gamma })\) if \(n \ge 4\).

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Acknowledgements

Part of the work was done while the third author was visiting Southwest university in China, which he thanks for the hospitality and financial support. The first and second authors are partially supported financially by NSFC (Nos. 11501468, 11501469). The third author is partially supported by the research fund of Hanyang University (HY-2017).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Wenjing Chen & Shengbing Deng

  2. Department of Mathematics, College of Natural Sciences, Hanyang University, 222 Wangsimni-ro Seongdong-gu, Seoul, 04763, Republic of Korea

    Seunghyeok Kim

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  1. Wenjing Chen
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Correspondence to Seunghyeok Kim.

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Communicated by P. Rabinowitz.

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Chen, W., Deng, S. & Kim, S. Clustered solutions to low-order perturbations of fractional Yamabe equations. Calc. Var. 56, 160 (2017). https://doi.org/10.1007/s00526-017-1253-2

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