Abstract
We study an elliptic problem involving critical growth in a strip, satisfying the periodic boundary condition. As a consequence, we prove that the prescribed scalar curvature problem in \({\mathbb {R}}^N\) has solutions which are periodic in some variables, if the scalar curvature K(y) is periodic.
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References
Ambrosetti, A., Azorero, G., Peral, I.: Perturbation of \(-\Delta u-u^{\frac{N+2}{N-2}}=0,\) the scalar curvature problem in \({\mathbb{R} }^N\) and related topics. J. Funct. Anal. 165, 117–149 (1999)
Bahri, A.: Critical points at infinity in some variational problems, Research Notes in Mathematics, vol. 82. Longman-Pitman, USA (1989)
Chang, S.Y.A., Yang, P.: A perturbation result in prescribing scalar curvature on \(S^n\). Duke Math. J. 64, 27–69 (1991)
Chang, S.Y.A., Gursky, M., Yang, P.C.: Prescribing scalar curvature on \(S^2\) and \(S^3\). Calc. Var. Partial Differ. Equ. 1, 205–229 (1993)
Chen, C.-C., Lin, C.-S.: Estimates of the conformal scalar curvature equation via the method of moving planes. Comm. Pure Appl. Math. 50, 971–1017 (1997)
Chen, C.-C., Lin, C.-S.: Prescribing scalar curvature on \(S^N\). I. A priori estimates. J. Differential Geom. 57, 67–171 (2001)
Chen, C.C., Lin, C.S.: Estimate of the conformal scalar curvature equation via the method of moving planes, II. J. Differ. Geom. 49, 115–178 (1998)
Deng, Y., Lin, C.-S., Yan, S.: On the prescribed scalar curvature problem in \({\mathbb{R} }^N\), local uniqueness and periodicity. J. Math. Pures Appl. 104, 1013–1044 (2015)
Guo, Y., Peng, S., Yan, S.: Local uniqueness and periodicity induced by concentration. Proc. Lond. Math. Soc. (3) 114, 1005–1043 (2017)
Li, Y.Y.: On \(-\Delta u=K(x)u^5\) in \({\mathbb{R} }^3\). Comm. Pure Appl. Math. 46, 303–340 (1993)
Li, Y.Y.: Prescribing scalar curvature on \(S^3, S^4\) and related problems. J. Funct. Anal. 118, 43–118 (1993)
Li, Y.Y.: Prescribing scalar curvature on \(S^n\) and related problems, Part I. J. Differ. Equ. 120, 319–410 (1995)
Li, Y.Y.: Prescribed scalar curvature on \(S^n\) and related problems II, Existence and comapctness. Comm. Pure Appl. Math. 49, 541–597 (1996)
Li, Y.Y., Wei, J., Xu, H.: Multi-bump solutions of \(-\Delta u=K(x)u^{\frac{n+2}{n-2}}\) on lattices in \(\mathbb{R} ^n\). J. Reine Angew. Math. 743, 163–211 (2018)
Li, Y., Ni, W.-M.: On the conformal scalar curvature equation in \({\mathbb{R} }^N\). Duke Math. J. 57, 859–924 (1988)
Rey, O.: The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal. 89(1), 1–52 (1990)
Schoen, R., Zhang, D.: Prescribed calar curvature problem on the \(n-\)sphere. Calc. Var. Partial Diff. Equ. 4, 1–25 (1996)
Wei, J., Yan, S.: Infinitely many solutions for the prescribed scalar curcature problem on \(S^N,\). J. Funct. Anal. 258, 3048–3081 (2010)
Yan, S.: Concentration of solutions for the scalar curvature equation on \({\mathbb{R} }^N,\). J. Differ. Equ. 163, 239–264 (2000)
Acknowledgements
Guo is partially supported by NSFC (12271283 12031015). Yan is partially supported by NSFC (12171184).
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Guo, Y., Yan, S. An elliptic problem with periodic boundary condition involving critical growth. Math. Ann. 388, 795–830 (2024). https://doi.org/10.1007/s00208-022-02550-1
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DOI: https://doi.org/10.1007/s00208-022-02550-1