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Classification of solutions for some elliptic system. (English) Zbl 1491.35179

Summary: In this paper, we classify the solution of the following elliptic system \[ \begin{cases} -\Delta u(x) = e^{3v(x)}, & x\in\mathbb{R}^4, \\ (-\Delta)^2v(x) = u(x)^4, & x\in\mathbb{R}^4. \end{cases} \] Under some assumptions, we will show that the solution has the following form \[ u(x)=\frac{C_1(\varepsilon)}{\varepsilon^2 + |x - x_0|^2},\quad v(x) = \ln\frac{C_2(\varepsilon)}{\varepsilon^2+|x-x_0|^2}, \] where \(C_1\), \(C_2\) are two positive constants depending only on \(\varepsilon\) and \(x_0\) is a fixed point in \(\mathbb{R}^4\).

MSC:

35J48 Higher-order elliptic systems
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
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[1] Brezis, H.; Merle, F., Uniform estimates and blow-up behavior for solutions of \(-\Delta u=V(x)e^u\) in two dimensions, Commun. Partial Differ. Equ., 16, 1223-1253 (1991) · Zbl 0746.35006 · doi:10.1080/03605309108820797
[2] Caffarelli, L.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear equations with critical Sobolev growth, Commun. Pure Appl. Math., 42, 271-297 (1989) · Zbl 0702.35085 · doi:10.1002/cpa.3160420304
[3] Chen, W.; Fang, Y.; Li, C., Super poly-harmonic property of solutions for Navier boundary problems on a half space, J. Funct. Anal., 265, 1522-1555 (2013) · Zbl 1288.35230 · doi:10.1016/j.jfa.2013.06.010
[4] Chen, W.; Fang, Y.; Yang, R., Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274, 167-198 (2015) · Zbl 1372.35332 · doi:10.1016/j.aim.2014.12.013
[5] Chen, W.; Li, C., Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63, 615-622 (1991) · Zbl 0768.35025 · doi:10.1215/S0012-7094-91-06325-8
[6] Chen, W.; Li, C.; Li, Y., A direct method of moving planes for the fractional Laplacian, Adv. Math., 308, 404-437 (2017) · Zbl 1362.35320 · doi:10.1016/j.aim.2016.11.038
[7] Chen, W.; Li, C.; Ou, B., Classification of solutions for a system of integral equations, Commun. Partial Differ. Equ., 30, 59-65 (2005) · Zbl 1073.45005 · doi:10.1081/PDE-200044445
[8] Chen, W.; Li, C.; Ou, B., Classification of solutions for an integral equation, Commn. Pure Appl. Math., 59, 330-343 (2006) · Zbl 1093.45001 · doi:10.1002/cpa.20116
[9] Chen, W.; Li, Y.; Zhang, R., A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272, 4131-4157 (2017) · Zbl 1431.35225 · doi:10.1016/j.jfa.2017.02.022
[10] Dai, W.; Qin, G., Classification of nonnegative classical solutions to third-order equations, Adv. Math., 328, 822-857 (2018) · Zbl 1429.35198 · doi:10.1016/j.aim.2018.02.016
[11] Gidas, B.; Ni, W.; Nirenberg, L., Symmetry and related properties via the maximum principle, Commun. Math. Phy., 68, 209-243 (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125
[12] Li, Y., Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6, 153-180 (2004) · Zbl 1075.45006 · doi:10.4171/JEMS/6
[13] Li, Y.; Zhu, M., Uniqueness theorems through the method of moving spheres, Duke Math. J., 80, 383-417 (1995) · Zbl 0846.35050 · doi:10.1215/S0012-7094-95-08016-8
[14] Lin, C., A classification of solutions of a conformally invariant fourth order equation in \(\mathbb{R}^n\), Comment. Math. Helv., 73, 206-231 (1998) · Zbl 0933.35057 · doi:10.1007/s000140050052
[15] Ngo, Q., Ye, D.: Existence and non-existence results for the higher order Hardy-Henon equation revisited. J de Mathématiques Pures et Appliquées, in press.
[16] Wei, J.; Xu, X., Classification of solutions of higher order conformally invariant equations, Math. Ann., 313, 207-228 (1999) · Zbl 0940.35082 · doi:10.1007/s002080050258
[17] Xu, X., Classification of solutions of certain fourth-order nonlinear elliptic equations in \(\mathbb{R}^4\), Pac. J. Math., 225, 361-378 (2006) · Zbl 1136.35039 · doi:10.2140/pjm.2006.225.361
[18] Xu, X., Exact solutions of nonlinear conformally invariant integral equations in \(\mathbb{R}^3\), Adv. Math., 194, 485-503 (2005) · Zbl 1073.45003 · doi:10.1016/j.aim.2004.07.004
[19] Zhu, N., Classification of solutions of a conformally invariant third order equation in \(\mathbb{R}^3\), Commun. Partial Differ. Equ., 29, 1755-1782 (2004) · Zbl 1129.35351 · doi:10.1081/PDE-200040197
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