Non-degeneracy for the critical Lane–Emden system

Frank, Rupert; Kim, Seunghyeok; Pistoia, Angela

doi:10.1090/proc/15217

Non-degeneracy for the critical Lane–Emden system
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by Rupert L. Frank, Seunghyeok Kim and Angela Pistoia

Proc. Amer. Math. Soc. 149 (2021), 265-278

DOI: https://doi.org/10.1090/proc/15217

Published electronically: October 16, 2020

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Abstract:

We prove the non-degeneracy for the critical Lane–Emden system \begin{equation*} -\Delta U = V^p,\quad -\Delta V = U^q,\quad U, V > 0 \quad \text {in } \mathbb {R}^N \end{equation*} for all $N \ge 3$ and $p,q > 0$ such that $\frac {1}{p+1} + \frac {1}{q+1} = \frac {N-2}{N}$. We show that all solutions to the linearized system around a ground state must arise from the symmetries of the critical Lane–Emden system provided that they belong to the corresponding energy space or they tend to zero at infinity.

References

A. Alvino, P.-L. Lions, and G. Trombetti, A remark on comparison results via symmetrization, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), no. 1-2, 37–48. MR 837159, DOI 10.1017/S0308210500014475
Shibing Chen, Rupert L. Frank, and Tobias Weth, Remainder terms in the fractional Sobolev inequality, Indiana Univ. Math. J. 62 (2013), no. 4, 1381–1397. MR 3179693, DOI 10.1512/iumj.2013.62.5065
Gabriele Bianchi and Henrik Egnell, A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), no. 1, 18–24. MR 1124290, DOI 10.1016/0022-1236(91)90099-Q
Josephus Hulshof and Robertus C. A. M. van der Vorst, Asymptotic behaviour of ground states, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2423–2431. MR 1363170, DOI 10.1090/S0002-9939-96-03669-6
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. MR 834360, DOI 10.4171/RMI/6
Guozhen Lu and Juncheng Wei, On a Sobolev inequality with remainder terms, Proc. Amer. Math. Soc. 128 (2000), no. 1, 75–84. MR 1694339, DOI 10.1090/S0002-9939-99-05497-0
Olivier Rey, The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), no. 1, 1–52. MR 1040954, DOI 10.1016/0022-1236(90)90002-3
James Serrin and Henghui Zou, Existence of positive entire solutions of elliptic Hamiltonian systems, Comm. Partial Differential Equations 23 (1998), no. 3-4, 577–599. MR 1620581, DOI 10.1080/03605309808821356
Xu Jia Wang, Sharp constant in a Sobolev inequality, Nonlinear Anal. 20 (1993), no. 3, 261–268. MR 1202203, DOI 10.1016/0362-546X(93)90162-L