Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case adding singular sources. I. (English) Zbl 1436.35148
Summary: We consider the existence of singular limit solutions for a nonlinear elliptic system of Liouville type in some general cases with singular source terms and Dirichlet boundary conditions. We use the nonlinear domain decomposition method.
MSC:
| 35J57 | Boundary value problems for second-order elliptic systems |
| 35B25 | Singular perturbations in context of PDEs |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
References:
| [1] | Bahrouni, A.; Ounaies, H.; Radulescu, V. D., Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh Sect. A, 145, 3, 445-465 (2015) · Zbl 1326.35329 |
| [2] | Bahrouni, A.; Ounaies, H.; Radulescu, V. D., Bound state solutions of sublinear Schrödinger equations with lack of compactness, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 113, 2, 1191-1210 (2019) · Zbl 1429.35079 |
| [3] | Baraket, S.; Bazarbacha, I., Singular limiting solutions for \(2\)-dimensional semilinear elliptic system of Liouville type adding singular sources terms given by Dirac masses, Complex Var. Elliptic Equ., 63, 1, 37-67 (2018) · Zbl 1393.35034 |
| [4] | Baraket, S.; Ben Omrane, I.; Ouni, T.; Trabelsi, N., Singular limits solutions for \(2\)-dimensional elliptic problem with exponentially dominated nonlinearity and singular data, Commun. Contemp. Math., 13, 04, 697-725 (2011) · Zbl 1231.35075 |
| [5] | Baraket, S.; Dammak, M.; Ouni, T.; Pacard, F., Construction of Singular limits for a \(4\)-dimensional semilinear elliptic problem with exponential nonlinearity, Ann. Inst. Henri Poincaré - AN, 24, 875-895 (2007) · Zbl 1132.35038 |
| [6] | Baraket, S.; Ouni, T., Singular limits solution for 2-dimensional elliptic problems involving exponential nonlinearities with sub-quadratic convection nonlinear gradient terms and singular weights, Adv. Nonlinear Anal., 3, 1, 69-88 (2014) · Zbl 1303.35025 |
| [7] | Baraket, S.; Pacard, F., Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations, 6, 1-38 (1998) · Zbl 0890.35047 |
| [8] | Baraket, S.; Sâanouni, S.; Trabelsi, N., Singular limit solutions for a \(2\)-dimensional semilinear elliptic system of Liouville type in some general case, Discrete Contin. Dyn. Syst., 40, 2, 1013-1063 (2020) · Zbl 1433.35071 |
| [9] | Baraket, S.; Ye, D., Singular limit solutions for two-dimentional elliptic problems with exponentionally dominated nonlinearity, Chinese Ann. Math. Ser. B, 22, 287-296 (2001) · Zbl 1009.35033 |
| [10] | Bartolucci, D.; Chen, C. C.; Lin, C. S.; Tarantello, G., Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations, 29, 7-8 (2004), 1241-1265 · Zbl 1062.35146 |
| [11] | Bonnett, W. H., Magntically self-focussing streams, Phys. Rev., 45, 890-897 (1934) |
| [12] | Chanillo, S.; Kiessling, M. K.H., Conformaly invariant systems of nonlinear PDE of Liouville type, Geom. Funct. Anal., 5, 6, 924-947 (1995) · Zbl 0858.35035 |
| [13] | Chen, W.; Li, C., Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63, 615-622 (1991) · Zbl 0768.35025 |
| [14] | Chen, W.; Li, C., Mean field equations of liouville type with singular data: sharper estimates, Discrete Contin. Dyn. Syst., 28, 43, 1237-1272 (2010) · Zbl 1211.35263 |
| [15] | Cirstea, F.; Radulescu, V., Entire solutions blowing up at infinity for semilinear elliptic systems, J. Math. Pures Appl., 81, 827-846 (2002) · Zbl 1112.35063 |
| [16] | Del Pino, M.; Kowalczyk, M.; Musso, M., Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24, 47-81 (2005) · Zbl 1088.35067 |
| [17] | Esposito, P., Blow up solutions for a Liouville equation with singular data, SIAM J. Math. Anal., 36, 4, 1310-1345 (2005) · Zbl 1162.35350 |
| [18] | Esposito, P.; Grossi, M.; Pistoia, A., On the existence of blowing-up solutions for a mean field equation, Ann. Inst. Henri Poincaré, 22, 227-257 (2005) · Zbl 1129.35376 |
| [19] | Ghergu, M.; Radulescu, V., (Singular Elliptic Problems Bifurcation and Asymptotic Analysis. Singular Elliptic Problems Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and its Applications, vol. 37 (2008), Oxford University Press) · Zbl 1159.35030 |
| [20] | Kiessling, M. K.H.; Lebowitz, J. L., Dissipative stationary plasmas; Kinetic modeling bennett’s pinch and generalizations, Phys. Plasmas, 1, 1841-1849 (1994) |
| [21] | Liouville, J., Sur l’équation aux différences partielles \(\partial^2 \log \frac{ \lambda}{ \partial u \partial v} \pm \frac{ \lambda}{ 2} a^2 = 0\), J. Math., 18, 17-72 (1853) |
| [22] | Nolasco, M.; Tarantello, G., Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. Partial Differential Equations, 9, 31-91 (1999) · Zbl 0951.58030 |
| [23] | Prajapat, J.; Tarantello, G., On a class of elliptic problems in R2: symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A, 131, 967-985 (2001) · Zbl 1009.35018 |
| [24] | Ricciardi, T.; Tarantello, G., Vortices in the Maxwell-Chern-Simons theory, Comm. Pure Appl. Math., 53, 811-851 (2000) · Zbl 1029.35207 |
| [25] | Struwe, M.; Tarantello, G., On multivortex solutions in Chern-Simons gauge theory, Boll. UMI, 8, 109-121 (1998) · Zbl 0912.58046 |
| [26] | Suzuki, T., Two dimensional Emden-Fowler equation with exponential nonlinearity, (Nonlinear Diffusion Equations and their Equilibrium Statesd, Vol. 3 (1992), Birkäuser), 493-512 · Zbl 0792.35061 |
| [27] | Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. Phys., 37, 3769-3796 (1996) · Zbl 0863.58081 |
| [28] | Trabelsi, M.; Trabelsi, N., Singular limits solutions for a 2-dimensional semilinear elliptic system of Liouville type, Adv. Nonlinear Anal., 5, 4, 315-329 (2016) · Zbl 1357.35145 |
| [29] | Yang, Y., Solitons in Field Theory and Nonlinear Analysis (2001), Springer-Verlag: Springer-Verlag New York · Zbl 0982.35003 |
| [30] | Ye, D., Une remarque sur le comportement asymptotique des solutions de \(- \Delta u = \lambda f ( u )\), C. R. Acad. Sci. Paris I, 325, 1279-1282 (1997) · Zbl 0895.35014 |
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.
