Uncountable sets of finite energy solutions for semilinear elliptic problems in exterior domains. (English) Zbl 1418.35170
Summary: In this article we discuss conditions suitable for the existence of asymptotically vanishing positive solutions for the following semilinear elliptic problem \(\Delta u(x) + f(x,u(x))+ g(x) x \cdot \nabla u(x) =0\), where \(x \in \mathbb{R}^n\) and \(||x|| > R\). The main result of our investigation is the construction of uncountable sets of minimal solutions which have finite energy in a neighbourhood of infinity. We apply an iteration scheme based on the subsolution and supersolution method. Our approach allows us to consider sublinear as well as superlinear problems without radial symmetry.
MSC:
| 35J61 | Semilinear elliptic equations |
References:
| [1] | Alarcón, S.; Quaas, A., Large number of fast decay ground states to Matukuma-type equations, J. Differential Equations, 248, 866-892 (2010) · Zbl 1187.35077 |
| [2] | Auchmuty, G., Finite energy solutions of self-adjoint elliptic mixed boundary value problems, Math. Methods Appl. Sci., 33, 12, 1446-1462 (2010) · Zbl 1197.35089 |
| [3] | Constantin, A., Existence of positive solutions of quasilinear elliptic equations, Bull. Aust. Math. Soc., 54, 147-154 (1996) · Zbl 0878.35040 |
| [4] | Constantin, A., Positive solutions of quasilinear elliptic equations, J. Math. Anal. Appl., 213, 334-339 (1997) · Zbl 0891.35033 |
| [5] | Constantin, A., On the existence of positive solutions of second order differential equations, Ann. Mat. Pura Appl., 184, 131-138 (2005) · Zbl 1223.34041 |
| [6] | Dat, C. T.; Verbitsky, I., Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms, Calc. Var. Partial Differential Equations, 52, 3-4, 529-546 (2015) · Zbl 1319.35063 |
| [7] | Deng, J., Bounded positive solutions of semilinear elliptic equations, J. Math. Anal. Appl., 336, 1395-1405 (2007) · Zbl 1152.35367 |
| [8] | Deng, J., Existence of bounded positive solutions of semilinear elliptic equations, Nonlinear Anal., 68, 3697-3706 (2008) · Zbl 1143.35036 |
| [9] | Djebali, S.; Orpel, A., A note on positive evanescent solutions for a certain class of elliptic problems, J. Math. Anal. Appl., 353, 215-223 (2009) · Zbl 1165.35015 |
| [10] | Djebali, S.; Orpel, A., Continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains, Nonlinear Anal., 73, 660-672 (2010) · Zbl 1194.35165 |
| [11] | Djebali, S.; Moussaoui, T.; Mustafa, O. G., Positive evanescent solutions of nonlinear elliptic equations, J. Math. Anal. Appl., 333, 863-870 (2007) · Zbl 1155.35029 |
| [12] | Ehrnström, M., Positive solutions for second-order nonlinear differential equation, Nonlinear Anal., 64, 1608-1620 (2006) · Zbl 1101.34022 |
| [13] | Ehrnström, M., On radial solutions of certain semi-linear elliptic equations, Nonlinear Anal., 64, 1578-1586 (2006) · Zbl 1121.35007 |
| [14] | Ehrnström, M.; Mustafa, O. G., On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal., 67, 1147-1154 (2007) · Zbl 1165.35016 |
| [15] | Engel, H., Asymptotic results for finite energy solutions of semilinear elliptic equations, J. Differential Equations, 98, 34-56 (1992) · Zbl 0778.35009 |
| [16] | Felmer, P.; Quaas, A.; Tang, M., On the complex structure of positive solutions to Matukuma-type equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26, 869-887 (2009) · Zbl 1175.35051 |
| [17] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer-Verlag: Springer-Verlag Berlin-Heidelberg, New York-Tokyo · Zbl 0562.35001 |
| [18] | Li, Y., On the positive solutions of the Matukuma equation, Duke Math. J., 70, 575-589 (1993) · Zbl 0801.35024 |
| [19] | Li, Y.; Ni, W.-M., On the existence and symmetry properties of finite total mass solutions of the Matukuma equation, the Eddington equation and their generalizations, Arch. Ration. Mech. Anal., 108, 2, 175-194 (1989) · Zbl 0705.35039 |
| [20] | Matukuma, T., The Cosmos (1938), Iwanami Shoten: Iwanami Shoten Tokyo |
| [21] | Ni, W.-M.; Yotsutani, S., Semilinear elliptic equations of Matukuma-type and related topics, Jpn. J. Appl. Math., 5, 1-32 (1988) · Zbl 0655.35018 |
| [22] | Noussair, E. S.; Swanson, C. A., Positive solutions of quasilinear elliptic equations in exterior domains, J. Math. Anal. Appl., 75, 121-133 (1980) · Zbl 0452.35039 |
| [23] | Noussair, E. S.; Swanson, C. A., Asymptotics for semilinear elliptic systems, Canad. Math. Bull., 34, 4, 514-519 (1991) · Zbl 0762.35030 |
| [24] | Noussair, E. S.; Swanson, C. A., Multiple finite energy solutions of critical semilinear field equations, J. Math. Anal. Appl., 195, 278-293 (1995) · Zbl 0865.35045 |
| [25] | Noussair, E. S.; Swanson, C. A.; Jianfu, Yang, Positive finite energy solutions of critical semilinear elliptic problems, Canad. J. Math., 44, 1014-1029 (1992) · Zbl 0795.35033 |
| [26] | Orpel, A., Increasing sequences of positive evanescent solutions of nonlinear elliptic equations, J. Differential Equations, 259, 5, 1743-1756 (2015) · Zbl 1318.35034 |
| [27] | Orpel, A., Minimal positive solutions for systems of semilinear elliptic equations, Electron. J. Qual. Theory Differ. Equ., 39 (2017) · Zbl 1413.35179 |
| [28] | Wang, B.; Zhang, Z., The radial positive solutions of the Matukuma equation in higher dimensional space: singular solution, J. Differential Equations, 253, 12, 3232-3265 (2012) · Zbl 1275.34017 |
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.
