Blow-up of solutions of critical elliptic equations in three dimensions. (English) Zbl 07898588
Summary: We describe the asymptotic behavior of positive solutions \(u_\varepsilon\) of the equation \(-\Delta u + au = 3u^{5-\varepsilon}\) in \(\Omega \subset \mathbb{R}^3\) with a homogeneous Dirichlet boundary condition. The function \(a\) is assumed to be critical in the sense of E. Hebey and M. Vaugon [Math. Z. 237, No. 4, 737–767 (2001; Zbl 0992.58016)], and the functions \(u_\varepsilon\) are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of H. Brezis and L. A. Peletier [Partial differential equations and the calculus of variations. Essays in Honor of Ennio De Giorgi, 149–192 (1989; Zbl 0685.35013)]. Similar results are also obtained for solutions of the equation \(-\Delta u + (a+\varepsilon V) u = 3u^5\) in \(\Omega\).
MSC:
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
References:
| [1] | 10.1016/0022-0396(87)90156-2 · Zbl 0657.35058 · doi:10.1016/0022-0396(87)90156-2 |
| [2] | ; Aubin, Thierry, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom., 11, 4, 573, 1976 · Zbl 0371.46011 |
| [3] | 10.1002/cpa.3160410302 · Zbl 0649.35033 · doi:10.1002/cpa.3160410302 |
| [4] | ; Brendle, Simon; Marques, Fernando C., Blow-up phenomena for the Yamabe equation, II, J. Differential Geom., 81, 2, 225, 2009 · Zbl 1166.53025 |
| [5] | 10.1002/cpa.3160390704 · doi:10.1002/cpa.3160390704 |
| [6] | 10.2307/2044999 · Zbl 0526.46037 · doi:10.2307/2044999 |
| [7] | 10.1002/cpa.3160360405 · Zbl 0541.35029 · doi:10.1002/cpa.3160360405 |
| [8] | 10.1007/978-1-4615-9828-2_7 · doi:10.1007/978-1-4615-9828-2_7 |
| [9] | 10.1017/S0308210500031140 · Zbl 0662.35003 · doi:10.1017/S0308210500031140 |
| [10] | 10.1016/j.aim.2016.03.043 · Zbl 1342.35085 · doi:10.1016/j.aim.2016.03.043 |
| [11] | 10.1016/S0294-1449(02)00095-1 · Zbl 1011.35060 · doi:10.1016/S0294-1449(02)00095-1 |
| [12] | 10.4171/JEMS/225 · Zbl 1210.35105 · doi:10.4171/JEMS/225 |
| [13] | 10.1007/BF01158557 · Zbl 0686.58043 · doi:10.1007/BF01158557 |
| [14] | 10.1016/j.na.2003.10.012 · Zbl 1134.35045 · doi:10.1016/j.na.2003.10.012 |
| [15] | 10.3934/mine.2020007 · Zbl 1537.35031 · doi:10.3934/mine.2020007 |
| [16] | 10.1007/s00526-021-01929-3 · Zbl 1511.35123 · doi:10.1007/s00526-021-01929-3 |
| [17] | 10.1007/s00209-004-0755-8 · Zbl 1122.35087 · doi:10.1007/s00209-004-0755-8 |
| [18] | 10.1016/S0294-1449(16)30270-0 · Zbl 0729.35014 · doi:10.1016/S0294-1449(16)30270-0 |
| [19] | 10.1016/j.anihpc.2007.03.006 · Zbl 1154.45004 · doi:10.1016/j.anihpc.2007.03.006 |
| [20] | 10.4171/134 · Zbl 1305.58001 · doi:10.4171/134 |
| [21] | 10.1007/PL00004889 · doi:10.1007/PL00004889 |
| [22] | ; Khuri, M. A.; Marques, F. C.; Schoen, R. M., A compactness theorem for the Yamabe problem, J. Differential Geom., 81, 1, 143, 2009 · Zbl 1162.53029 |
| [23] | 10.4171/aihpc/95 · doi:10.4171/aihpc/95 |
| [24] | 10.1090/gsm/014 · doi:10.1090/gsm/014 |
| [25] | 10.4171/RMI/6 · Zbl 0704.49005 · doi:10.4171/RMI/6 |
| [26] | 10.1093/imrn/rnaa021 · Zbl 1493.58007 · doi:10.1093/imrn/rnaa021 |
| [27] | 10.1007/s11118-013-9340-2 · Zbl 1286.35084 · doi:10.1007/s11118-013-9340-2 |
| [28] | 10.1016/j.jde.2018.01.046 · Zbl 1394.35145 · doi:10.1016/j.jde.2018.01.046 |
| [29] | 10.1016/j.matpur.2004.02.007 · Zbl 1130.35040 · doi:10.1016/j.matpur.2004.02.007 |
| [30] | 10.1007/978-3-0348-0373-1_5 · Zbl 1283.35022 · doi:10.1007/978-3-0348-0373-1_5 |
| [31] | 10.1007/BF01168364 · Zbl 0708.35032 · doi:10.1007/BF01168364 |
| [32] | 10.1016/0022-1236(90)90002-3 · Zbl 0786.35059 · doi:10.1016/0022-1236(90)90002-3 |
| [33] | 10.1137/0121004 · Zbl 0201.38704 · doi:10.1137/0121004 |
| [34] | 10.1007/BF01174186 · Zbl 0535.35025 · doi:10.1007/BF01174186 |
| [35] | 10.1007/BF02418013 · Zbl 0353.46018 · doi:10.1007/BF02418013 |
| [36] | 10.7146/math.scand.a-10841 · Zbl 0164.13101 · doi:10.7146/math.scand.a-10841 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
