Chen, Xu-Yan; Matano, Hiroshi; Véron, Laurent Anisotropic singularities of solutions of nonlinear elliptic equations in \({\mathbb{R}}^ 2\). (English) Zbl 0687.35020 J. Funct. Anal. 83, No. 1, 50-97 (1989). The authors consider the equation \((1)\quad \Delta u=| u|^{q- 1}u,\) \((q>1)\) in \(\Omega \subset {\mathbb{R}}^ 2\). The main subject of this paper is to classify all possible isolated singularities of solutions of (1). They use an infinite-dimensional dynamical systems theory to prove the existence of various different types of global singular solutions. Reviewer: Chuanfang Wang Cited in 27 Documents MSC: 35B99 Qualitative properties of solutions to partial differential equations 35J60 Nonlinear elliptic equations 70G99 General models, approaches, and methods in mechanics of particles and systems Keywords:singularities; semilinear Laplace equation PDFBibTeX XMLCite \textit{X.-Y. Chen} et al., J. Funct. Anal. 83, No. 1, 50--97 (1989; Zbl 0687.35020) Full Text: DOI References: [1] Aronszajn, N., A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl., 36, 235-249 (1957) · Zbl 0084.30402 [2] Aviles, P., Local behaviour of solutions of some elliptic equations, Comm. Math. Phys., 108, 177-192 (1987) · Zbl 0617.35040 [3] Berestycki, H., Thése de Doctorat d’Etat. Univ. Paris 6 (1980) [4] Brezis, H.; Lieb, E. H., Long range atomic potentials in Thomas-Fermi theory, Comm. Math. Phys., 65, 231-246 (1979) · Zbl 0416.35066 [5] Chafee, N.; Infante, E. F., A bifurcation problem for a nonlinear partial differential equation of parabolic type, Appl. Anal., 4, 17-37 (1974) · Zbl 0296.35046 [6] Chen, X.-Y, Uniqueness of the ω-limit point of solutions of a semilinear heat equation on the circle, (Proc. Japan Acad. Ser. A Math. Sci., 62 (1986)), 335-337 · Zbl 0641.35028 [8] Chen, X.-Y; Matano, H.; Véron, L., Singularités anisotropes d’équations elliptiques semi-linéaires dans le plan, C. R. Acad. Sci. Paris Sér. I Math., 303, 963-966 (1986) · Zbl 0637.35033 [9] Fowler, R. H., Further studies on Emden’s and similar differential equations, Quart. J. Math., 2, 259-288 (1931) · Zbl 0003.23502 [10] Gidas, B.; Spruck, J., Global and local behaviour of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34, 525-598 (1981) · Zbl 0465.35003 [11] Gilbarg, D.; Trüdinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0691.35001 [12] Loewner, C.; Nirenberg, L., Partial differential equations invariant under conformal or projective transformations, (Contributions to Analysis (1974), Academic Press: Academic Press Orlando, FL), 245-272 [13] Matano, H., Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18, 221-227 (1978) · Zbl 0387.35008 [14] Matano, H., Existence of nontrivial unstable sets for equilibriums of strongly orderpreserving systems, J. Fac. Sci. Univ. Tokyo, 30, 645-673 (1983) · Zbl 0545.35042 [17] Vazquez, J. L.; Véron, L., Singularities of elliptic equations with an exponential nonlinearity, Math. Ann., 269, 119-135 (1984) · Zbl 0567.35034 [18] Vazquez, J. L.; Véron, L., Isolated singularities of some semilinear elliptic equations, J. Differential Equations, 60, 301-321 (1985) · Zbl 0549.35043 [19] Véron, L., Singular solutions of some nonlinear elliptic equations, Nonlinear Anal., 5, 225-242 (1981) · Zbl 0457.35031 [20] Véron, L., Comportement asymptotique des solutions d’équations elliptiques semilinéaires dans \(R^n\), Ann. Mat. Pura Appl., 127, 25-50 (1981) · Zbl 0467.35013 [21] Véron, L., Global behaviour and symmetry properties of singular solutions of nonlinear elliptic equations, Ann. Fac. Sci. Toulouse Math., 6, 1-31 (1984) · Zbl 0561.35031 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.